When a Monster's 'Likes' trait changes to your name, meaning they like you more than whatever was there previously.
The first game only has the Loyalty meter and the monster's obedience to indicate their affinity for you. Starting with 2, though, a Monster can like you so much that they'll want to play with you. It's really, really hard to say "No" to this, especially since they have a better chance of getting great results on their next drill because of how happy it makes them when you say "Yes."
After winning a major tournament, your monster will usually ask for something it likes to celebrate.
The intro to Monster Rancher 4 has the hero and his Monsters getting Rio a flower.
In Monster Rancher 4, Phayne once had a Mocchi. He had no parents, so it was his only friend. But he treated it more like a stuffed animal, never thinking about its feelings or needs. One day, when his Mocchi didn't even want to be outside, they were attacked by a stray monster. The Mocchi lunged for it, and they both fell into the sea. After that, Phayne was unable to raise Mocchis. It has a happy ending, though. After you beat Geras, Phayne's Mocchi returns, and is said to be living comfortably with other Mocchis.
The first, second, and third Japanese opening themes are pretty heartwarming.
In episode 1, Genki has just met Mocchi when the latter is threatened by the Black Dino Squad. What does Genki do when their leader threatens to make the young monster a slave? Kick the Black Dino in the face. Big Brother Instinct established.
In the second episode, Genki and Holly unlocked a Spot Dino, and gave it to an elderly couple they'd befriended earlier who wished for a Monster to love.
In return, the couple gave Genki shoes to use alongside his rollerblades, and a cloak. These were worn throughout the series, and it all started here.
Also in the second episode, Mocchi ran away because he thought Genki didn't love him. Genki went out to look for him and soon got lost. When bad guys cornered Genki and asked him where Mocchi was, Genki wouldn't tell. Mocchi then realized how much Genki loved him and ran out to save him.
Episode 5 "Tiger of the Wind", Genki and Holly bonding with Tiger after his pack is killed.
"The Courageous Seven" has the Searchers working as fast as they can to build a new dam before the one some Evil Hares damaged breaks, flooding a village. While they manage to construct one in time, the first dam breaks and it looks as if the new dam won't hold up. Cue Genki running towards it and holding back the falling dam by himself. His determination to not let Moo win inspires the other Searchers, and eventually the entire town, to join him in holding back the dam... and once everyone joins together, it holds.
After hearing Holly's tear-jerking story about her past in episode 8, Genki and the others sing a silly song to try and make her laugh.
Although the really heartwarming bit from that sequence is Golem gently picking Holly up, whereupon she stands up on his palm, throws her head back, eyes closed and a big happy smile on her face, and says "From now on, the rain will make me happy".
One from "Pixie's Defeat" where Genki and Mocchi have to split up to help save the others. Mocchi is reluctant to leave, but Genki insists.
Genki: Mocchi, listen. If I or the others don't make it back, someone has to find the Phoenix...
Mocchi: ...I understand, chi. [hugs him]
In "Suezo's Secret Weapon," Suezo is depressed because he can't catch up to the others' abilities. Genki and Mocchi try and train him in telepathy, but it fails, so Holly gives him a pep talk to cheer him up. This pays off when he learns how to teleport, saving them from going down a waterfall, and everyone is genuinely impressed.
"My Name Is Pixie" has Genki and Pixie saving each other throughout the episode, eventually coming to an understanding.
Holly resolving to stick by Suezo in "Battle With the Big Bad Four".
"Amusement Park Ruins" has everyone's delight at the old amusement park, especially when Pixie's lightning gets it working again.
Episode 24 "Undine's Lake" Golem befriending and falling in love with Undine, referring to them as soulmates.
Tiger and Grey Wolf's time as puppies.
Tiger tried to act like an Aloof Big Brother so that Grey Wolf would toughen up, but he ends up crumbling a few seconds after his little brother starts crying. Their flashback ends with Grey Wolf happily snuggling into Tiger and calling him the best big brother ever.
The entirety of "Holly's Happy Birthday", where Holly gets a surprise birthday party from Genki, Mocchi, Golem, Suezo, Tiger and Hare.
Genki and Mocchi apologizing to each other in "Renocraft, Slayer in the Sand".
Captain Astro. He tells Genki and Holly of how the ancient people ran out of resources and forced the Astros to look for more, working them well past their breaking points. Many of the Astros died, and now the Captain accuses humans of being evil and destructive. Genki tells him that not all humans are like that, but he won't listen... until Genki risks everything to save his life as they're about to go down a waterfall. With Suezo's grip on them weakening, Astro shoots his rocket fist to save Genki, finally realizing not all humans are bad. Despite the fall, Captain Astro meets with his troops and tells them to call off their war on the humans, asking that they try to enjoy their freedom... and then dies.
Genki was absolutely heartbroken over it, only jumping up happily after being prompted to. But as he was crying, one of the other Astros wiped away his tears.
"Tears" has the reveal that everyone Genki has met and helped on his journey is fighting Moo's forces.
The second season finale was this combined with a Tear Jerker.
Genki's reunion with everyone in Season 3.
Tiger and Grey Wolf's reunion in Season 3.
Because they're about to face each other in a Monster fighting tournament, Tiger purposely acts like a jerk to Grey Wolf so that the latter wouldn't be held back by his emotions. However, Tiger ends up being the one to instinctively hold himself back when he's unable to use his most powerful attack while he has Grey Wolf cornered, not wanting to go all out on his younger brother.
Then when Grey Wolf gets caught off guard by another monster attacking him from the side, Tiger instantly leaps in front of him to protect him with the lightning attack he was previously unable to use.
The third season finale managed to be this, without the Tear Jerker, as Holly's father's memories are restored and Moo's evil is defeated once and for all.
Episode 73 has Mum Mew screaming that she likes herself just as she is when Moo's soul starts to consume her and the others. After an entire season of buying exercise gadgets and hating being called old, when her life is on the line Mum Mew accepts herself and her body image.
The Japanese version of the last episode shows an epilogue English viewers missed out on; Genki being able to visit Monster Rancher world whenever he wants, and Mocchi somehow having the power to appear in the "Real" world, and that only other children can see him. It also includes shots of the others in peaceful lives.