The Amazing Race, being a competition where you only have to beat one other team to the Pit Stop to survive, and bad luck can hit anyone at any time, has featured a large number of Elimination Houdini's.
Lenny & Karyn, Season 1, beat out several stronger teams despite finishing in the bottom 3 in every leg except the first one. They barely avoided elimination on several occasions due to a team getting lost in the desert, another team not having the proper currency to pay their taxi, and a non-elimination leg.
Steve & Dave, the air traffic controllers from Season 4. After an injury in the first leg they refused to run, yet lasted four more legs thanks to a Fast Forward and major blunders by other teams. On their penultimate leg, the show even played the "sad elimination" music for their finish, only to subsequently cut to a forgotten team that had taken a wrong turn and fallen hopelessly behind.
Kami & Karli, Season 5, finished either last or second-to-last on 6 of the first 9 legs, making many blunders along the way (such as being the only team who didn't figure out how to use puzzle pieces and a map to find out the next destination on their own), before finally getting eliminated on leg 10.
Adam & Rebecca, Season 6, finished in the top half of the field only once on their way to making the Final 3, and that one leg they won was by Fast Forward. They made several mistakes along the way, barely avoiding elimination at least 4 times. In the Sri Lanka leg, they had fallen over an hour and a half behind behind every other team, only to be saved by arguably the most badly placed equalizer in the series (at the END of the leg).
Also from that season were Don & Mary Jean, who stumbled through 5 legs despite accidentally stealing another team's car in leg 2, wasting over 5 hours at the IKEA counting detour in leg 3, and choosing the slower, luck-based fishing Detour in leg 4 (and finishing almost 4 hours behind the first team). They were finally eliminated in 8th after getting lost at the beginning of the 5th leg, after which not even Lori & Bolo accidentally walking several miles could save them.
Meredith & Gretchen, Season 7, who were the first team to succeed while deliberately using the "Just beat one team" strategy, even though Gretchen got injured and they had their bags taken away on an early leg.
Ray & Yolanda, Season 9, appeared to be slow at everything, surviving mostly because Season 9 teams were extremely mistake prone. They finished second-to-last in 4 of the last 5 elimination legs, and at one point survived because another team Yielded the team right in front of them.
Despite winning five legs, BJ & Tyler went through a Houdini stage in the middle of Season 9. Over the course of four legs, they came in last on two different non-elimination legs, and survived a third time when another team got lost looking for the Pit Stop.
Lyn & Karlyn, Season 10, who were the first female team to make the Final 3 despite being out of shape and constantly struggling to keep up with the more fit teams. They didn't finish in the top half of the field until the 10th leg, and only got that far because of stupid decisions by their alliance-mates Erwin & Godwin.
Karlyn: We just won't die. Like roaches.
Eric & Danielle, from Season 11, were the first Elimination Houdinis to win the race. In the first 11 legs, they had one top 2 finish, and their five bottom 2 finishes without getting eliminated is tied for third most ever, trailing only Kami & Karli's six in Season 5 and Josh & Brent's seven in Season 21. They also had two legs where they finished more than twelve hours behind the lead team, and were only able to catch up to the leaders due to the most blatantly contrived equalizer in race history. And ultimately, they only won because of a task involving guessing their partner's responses to questions about the other teams. Aside from being hugely luck-based considering it was the final task, the favorites to win, Dustin & Kandice, claimed in a post-race interview that Dustin's initial response wouldn't go through.
Dan & Andrew from Season 13, who only made it to the Final 3 because they were repeatedly saved by non-elimination legs or another team making an even worse blunder than they did. They are probably the Race's worst offenders/best examples of this trope, and are widely credited as being the worst team to ever make the Final 3. Phil even told them at one point that he thought they were going to be the next team eliminated. They even acknowledged how bad they were, saying that there should be a YouTube compilation video of all their mistakes - and one *was* created. They at least are able to laugh at themselves.
Mark & Michael, Season 14, who had a knack for getting tragically lost on every single leg. They are the only team to ever get multiple penalties on two different legs, and two different penalties on the same task (for sabotaging the other teams, and getting outside help when the clue told them that was not allowed), yet they managed to barely escape elimination on at least three separate occasions. This includes another team getting U-Turned, a team not having enough time to make up their Speed Bump, and a non-elimination leg.
Brian & Ericka, Season 15, survived twice, despite huge deficits, because another team quit a task. Maria & Tiffany quit a Detour where they were physically unable to do either task, while Big Easy quit a Roadblock where he was unable to unscramble the name "Franz".
Brent & Caite, Season 16, went through dehydration, got lost several times, and constantly bickered, yet made it to the Final 3 over several stronger teams.
Cops Louie & Michael as well. Other than a stretch of three legs without an equalizer, they spent all their time at or near the back of the pack, barely avoiding elimination at least four times thanks to Jody and Shawne struggling with Roadblocks in consecutive legs, Carol & Brandy getting U-Turned, and a non-elimination leg.
Nick & Vicki, Season 17. They spent most of their time in the back of the pack, and rarely completed non-physical tasks on the first try. Yet despite all this, and Nick's constant verbal abuse towards Vicki, they finished 4th due to other teams making mistakes, and twice were saved by non-elimination legs. On the third-to-last leg, Nick even gave up and napped during the Detour and eventually dragged Vicki into accepting a 6-hour penalty to skip it, only to be saved by a non-elimination leg. Thankfully, it only turned out to delay the inevitable - their late start time on the following leg caused them to miss the flight the other racers took out of Hong Kong, and they never came close to catching up after that.
Season 17 also had mistake prone Chad & Stephanie, who at one point were saved despite finishing in last with a 30-minute penalty because another team had a longer penalty.
Season 18 had Kent & Vyxsin barely escape elimination several times, including after running probably the worst leg of any team ever. Leg 4 started with them driving in the wrong direction for several hours, and catching their mistake so late that they missed the production-arranged flight out of Japan; the usual equalizer that everyone else reached with hours to spare. They continued to panic and melt down throughout the leg, topping it off by leaving their Race pack behind on a gondola, losing their passports for the second time in the same leg. They got saved by the fact that it was a double leg, catching up at the next equalizer. Then, on the next leg, Kent fell off a ladder during a Roadblock, cracking his rib and spraining an ankle, yet the lasted five more legs after that.
Marcus & Amani, Season 19, were constantly making mistakes and putting themselves in the back of the pack and being forced to play catchup, surviving mostly due to other teams making even worse mistakes at the last minute. Marcus, despite being a former NFL tight end, got shown up on physical tasks by a 60-year-old man. And in the penultimate leg, despite not discovering the location of the Pit Stop for themselves, their cab driver got it from another team's cab driver, putting them in the Final 3.
Vanessa & Ralph, Season 20, several times got themselves dreadfully behind the lead pack, yet managed to survive because someone did just a little bit worse than them. They survived once when Elliot & Andrew struggled with the Detour for as long as they did, again when Joey "Fitness" & Danny wasted time on a Fast Forward they didn't get, then later almost blew a two-hour lead on Nary & Jamie on a leg without an equalizer.
Also from Season 20 were Mark & Bopper. While they were competent enough, they were consistently plagued by injuries, illness, and bad navigation (mostly in leg 4, where they somehow ended up about 6-9 hours behind the leaders). They were saved twice by non-elimination legs; and even benefited from great luck on their final leg - just not enough to make up the massive time lost on the previous leg (nobody else had claimed the Fast Forward ahead of them, and it was shaving their heads when Mark was already bald).
From Season 21 were eventual winners Josh & Brent, who set the franchise record for finishing in the bottom 2 for seven straight legs and surviving all seven legs (Their final leg victory was the first time since leg 4 where they finished above the bottom 2). They were saved once by a non-elimination leg, survived a leg where they fell more than 5 hours behind the lead four teams and had to take a 4-hour penalty after Brent was unable to complete the swimming Detour when James & Abba lost their passports and were unable to retrieve them, and again the next leg after that, in spite of their flight being over 2 hours behind the other teams' when Abbie & Ryan 1) had their earlier flight delayed and ended up arriving at Amsterdam the same time as them and 2) were U-Turned by Jaymes & James. If there was ever a team who suffered from unbelievably bad and unbelievably good luck at the same time, it's them.
Also Gary & Will. They survived two legs in a row consisting of massive blunders (including U-Turning a team ahead of them) and being ready to give up because the teams they beat out were derailed by bad drivers, before finally getting eliminated in leg 4.
Chuck & Wynona in Season 22. In three straight legs they were saved by a fake Pit Stop non-elimination, Jessica & John getting eliminated because they refused to use their Express Pass, and Dave & Connor quitting the race, which saved them despite finishing almost three hours behind the other teams. They were later eliminated due to a 30-minute penalty for not using their sleds on the cheese hill task, even though they were nearly saved by Meghan & Joey walking to the Pit Stop when a local gave them bad directions.
Max & Katie also qualify. After they got two other teams to quit a task with them in the first leg and take a four hour penalty, they were handily beaten by the other surviving team in the next leg, only staying in by having their penalty cut down by an Hours of Operation bunching point and two other teams' major screwups. In the next leg, they screwed up navigation so badly that they were only spared from last place by Chuck & Wynona screwing up worse. They were later saved by a non-elimination leg after being the last team to quit the longer end of a Detour, and in the leg after that, despite a Speed Bump, navigation issues, and having to pay a speeding ticket, they managed to survive again due to Pam & Winnie imploding on the Detour. However, they did end up winning three straight legs in Europe just before the finale.
Mona & Beth took up the Elimination Houdini mantle in the last third of the Race (when Chuck & Wynona were eliminated and Max & Katie started winning legs) as their ineptitude at navigation and mental tasks started to show, only making the Final Three due to a non-elimination and two other teams struggling with physical tasks. In the 10th leg, they managed to survive both a Speed Bump (one of the harder ones in the series) and a U-Turn, beating out Joey & Meghan because they had also been U-Turned.
Ally & Ashley from Season 23, who ran along at the back of the pack for most of the season, and finished five straight legs in second-to-last place.
Travis & Nicole were a unique case as, despite being established as the strongest team up until then, Nicole broke down at the Leg 9 and 11 Roadblocks and only stayed in the race because Amy helped her out both times.
In Season 24 there's Brendon & Rachel. While they finished in the top 2 for five of the legs, they've barely escaped elimination in every other leg thanks to other peoples mistakes (Joey & Meghan letting their cab go in the middle of nowhere) or being lucky at the right time (non-elimination, being right on a coin flip flight decision, John telling Rachel what she was doing wrong on a Roadblock).
Then there's Caroline & Jennifer. Though they spent most of the season in the middle of the pack, the two times they fell to last they were saved by non-elimination.
Zabrina & Joe Jer from Season 1. After they were non-eliminated in leg 7, they spent the next four legs in the bottom 2. They are able to salvage it and won the race.
Terri & Henry, Season 2. They were consistently second-to-last place on several legs. Also, they were spared by two consecutive non-elimination legs despite the huge time deficit between the other racers.
Tania & Ida, Season 3. They were also spared by two consecutive non-elimination legs. The only difference is that they were able get to the final three.
Natasha and Hussein, Season 4. They made several critical mistakes throughout the race such as going to the wrong boat on the first leg, they failed the Roadblocks twice in Malaysia and Australia and were forced to take the 4 hour penalties. Despite of that, they were spared due to the other racer's misfortunes and the leg in Australia being a double-length leg.
Henry & Jennifer (3), they were saved by 2 non-elimination legs, they were saved from elimination twice when one team got lost and another team got penalized for leaving their bags in the wrong area, stayed in the back of the pack consistently and always placed in the bottom 3 (except for 2 legs), yet they managed to last 7 legs.
Mo & Mos (1) came last on the first non-elimination leg, were saved from almost certain elimination thanks to Anne-Marie & Tracy quitting the same Roadblock they did, and survived the first leg in South Africa only thanks to karma coming back to bite Richard & Joey in the ass for being unbearable to every other team. They finally were eliminated on leg 5 even though fate seemed to give them every chance to survive, first when they got lucky with an Intersection (allowing them to catch up to Sam & Renae), then again when Chris & Anastasia lost their car keys (and even when, by a small miracle, they found the key, they were unable to open with without Sam & Renae's help).
After Mo & Mos were gone, Matt & Tom slipped into the roll. First, on leg 6, Matt burned through all his arrows on the Roadblock, and only Jeff's charity kept him from having to take a four-hour penalty. They then came in last at the midpoint of the Super Leg, then were the last to check in again at the next Pit Stop, only to be saved by Chris & Anastasia's penalty.
Lucy & Emilia, 2. Despite being outmatched by pretty much every other team, they managed to make it to 5th place due to being saved by the Salvage Pass in the first leg, and everyone else making huge mistakes afterwards. This includes teams taking 4 hour penalties, a team getting so lost that Grant eliminated them in the field while they were still in the middle for the leg, and a team missing an hours of operation equalizer and being broke on top of that.
Tim Sr. & Tim Jr. in Season 1, who were saved by both of the first two non-elimination legs, and managed to constantly make mistakes that sent them to the back of the pack. They managed to get to the Final 3 and win the Race.