If a human with the mark of handling variant race can cast animal friendship  on monstrosities with low enough INT, could they theoretically control the Tarrasque? Having the most powerful creature in the entire game on your side seems too good to be true. Is there anything I'm missing here?

Nevermind, the spell charms the target and the Tarrasque is immune to being charmed. Are there any other creatures this could work on?

Note that even if you could cast it on a tarrasque, it would NOT let you control it! Animal Friendship’s effect is “This spell lets you convince a beast that you mean it no harm.” Not mind control. It does not put the creature “on your side”. It also inflicts the charmed condition, which means it can’t attack you and you have advantage on social interaction with it. Still not “on your side!”

Not just that but Animal Friendship targets a Beast and a Tarrasque is a Monstrosity. So it would be an invalid target even if it weren't immune to charmed.

Overall, this seems to be a common misconception with many mental spells. Charm Person, Friends, Animal Friendship, swashbuckler's Panache, heck, even generic persuasion or animal handling checks get this treatment sometimes.

None of those are mind control. They have very specific effects, which are NOT "the party gets to tell the creature exactly what to do, and it listens".

That's Dominate Person and Dominate Monster. Anything less than those is going to have various limitations.

I would rule that the spell just fails. The tarrasque is not an animal; it's a force of nature. Casting animal friendship on a tarrasque has the same effect as casting it on a thunderstorm.

Have you tried the wish spell.