In the beginning of Episode One, Dog suggests getting Alyx and Gordon into the Citadel by tossing a car (with them in it) across a gap into the building.
Alyx: Well, Gordon, unless you have a better suggestion...he is a robot, he's done the math.
*Turns to Dog*
Alyx: You did do the math, right?
Dog: *Shakes head no*
In addition, if you listen to the developer commentary, you'll find out that this joke started because of a glitch in which Dog's head shook back and forth after Alyx mentioned him doing the math. The play testers loved it so much that Valve made it official.
"Now would be an excellent time for procreation! That is to say, in layman's terms, you should seriously consider doing your part for the revival of the species." Alyx: Uh, is Dr. Kleiner really telling everyone to... get busy?
Even funnier is when, in Episode Two's recap of One, they replay Dr. Kleiner saying "Now would be an excellent time for procreation!" while the city's on fire and Combine are running around shooting everyone.
You can derive great fun from throwing things at the officers in Half-Life 2.
Let's see: You've got standard objects like exploding and non-exploding barrels, but there are also things like tires and boxes. And if you're just feeling silly, you can toss paint cans at them!
You can also throw toilets at your enemies. You even get an achievement for it.
In Episode 2, where you have to help Greggs and Sheckley fight off the Antlions, after Alyx is revived by the Vortigunts and Alyx and Gordon are on their way riding up the lift, Greggs yells "Alright Gordon, be adequate!", to which Sheckley asks him what the hell that means. Greggs simply replies, "I don't know, sounds like something the Vortigaunts would say."'
Almost everything Griggs says counts as this. "I think we misunderestimated!"
When you enter the Victory Mine, you see these little immobile grubs all over. Because you can easily kill them for health globs and there's even an achievement for getting all them, you'll probably try and kill as many as possible. When you reach Griggs and Sheckly's outpost they say the Antlions don't make themselves too much of a problem so long as you don't mess with their grubs. Whoops.
In Episode 1, there is a moment where you have to crawl up the air ducts and temporarily leave Alyx behind; however, if you go too far down the wrong path you fall through, in the same room where you started from right in front of Alyx.
"Oh, back so soon?"
In Episode 2, when infiltrating a Combine-occupied warehouse to destroy an autocannon, there's a particularly amusing scene (even noted by the developers) where a fast-zombie torso is being chased Tom-and-Jerry style by a Combine soldier.
After the Combine teleporter explodes as Alyx and Gordon escape Nova Prospekt, they arrive at their destination safely... a week later. Dr. Kleiner is momentarily distracted by the physics questions this malfunction reveals: "Fascinating! We seem to have developed a very slow teleport!"
Alyx Face Palms when she realizes that Kleiner has totally forgotten the armed revolution going on outside.
Due to programming oversight, if you bring a (friendly) Turret into the teleport room in Entanglement, it will shoot Eli Vance as soon as he emerged from the prisoner pod, dropping him dead for good and stopping the scripted event short. Funny that Alyx would forget to remove her dad's name from the Turret's "enemy list" when she reprogrammed it.
"Go on across, Gordon. She's waiting for ya! You lucky dog you."
The first time you see D.O.G. in action. He gets hit by an APC which seems to take him out... until you see the tank driving backwards down the road with D.O.G. sitting on the hood pounding it with his fists. He then flips the APC over with his feet and tosses it into a group of hapless Combine soldiers.
In Episode 1, during the section where you need to shine your flashlight on enemies to have Alyx take them out, turn off your flashlight and there's a random chance of Alyx starting to make zombie noises. If you turn it on soon enough, she'll laugh and say "Gotcha.".
When the teleporter glitches out in "Red Letter Day" in HL2 and sends you all over City 17, at one point it ports you straight into Breen's office, startling him and making him recognize you. After a few more teleport spots, it briefly drops you back into Breen's office where he's talking to a Combine Advisor on a viewscreen about Freeman suddenly 'porting into his office.
The first time the player ends up in his office...
Dr. Breen: *Look up and recoils in surprise* What's the meaning of this? Who are you? How did you get in here?
The second time...
Dr. Breen: The man I saw, I'm all but certain it was...*turns to look*...Gordon Freeman.
After defeating the giant Antlion Guard in HL2 chapter "Sandtraps", 2 Vortigaunts are talking in their own language, then stop to apologize to Gordon, saying what they're doing is rude. They then promise to speak English when Gordon is present, unless they wish to insult him. They then immediately resume speaking Vortuguese.
Vortigaunts, the somewhat creepy yet incredibly awesome lighning-throwing aliens in Half-Life, apparently speak like stereotypical Southern ministers