By the book, the 5th edition Wish is single-handedly the most limited version of Wish to have ever existed. Before that, Wish was literally the most powerful spell because it had no limits, and 5th edition puts a serious kibosh on what you can do with it.
That said: generally "Wish" has three outcomes:
1: Compromise. The "perfect" result where a player wishes for something cool, but something reasonable and is willing to accept perhaps not the perfect incarnation of what they want, but an incarnation that makes them happy and doesn't break the game. This can be anything from an existing item, a minor stat boost, wishing to get some hard-to-reach item or travel to some exotic location, perhaps even revive an NPC that was regrettably lost in some previous part of the game. It's a situation where the DM doesn't need to screw with the player, and the player isn't trying to wreck the game. Works for everyone involved.
2: Anything. A situation where the DM is totally agreeable to anything the player wants (for whatever reason) and the player asks for the moon and the sun combined into their own personal universe filled with busty elf chicks and and endless supply of beer. This can be fun. It can sometimes work out for the game, but generally it comes back to bite everyone in the butt.
3: Trick E. DMson. A situation where the player asks for something, regardless of if it is reasonable or not and the DM takes it upon themselves to play mischievous genie. It can be fun, but more often than not it leads to resentment, because the DM purposefully gives the player what the player doesn't want in order to strip them of the power granted them by the wish spell. Lets say the player asks for a Flaming Sword, not an unreasonable request and a great chance for situation 1, but instead the DM decides to screw with the player and gives them a sentient homosexual sword (playing on the word "flaming"). The sword makes obnoxious gay-sex related jokes has a lisp that would make Queer-Eye shudder and is generally annoying until the player gets rid of it. Or lets say, the player wishes for immortality (again, not an unreasonable request given the Wish spell) and the DM turns him into stone (ala the Flesh to Stone spell). Very classic magical turnabout, but it just doesn't work in a cooperative gaming environment.
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As a DM, I ride somewhere between "anything" and "compromise". I'll generally give my players what they want and use their own desires to further the game. Wish to be a dragon? Now powerful dragon-hunters are out to kill you and the party! Wish for a flaming sword? Matter cannot be created or destroyed: your flaming sword is the very one that magically vanished from the hands of a secretive monastary and now ninjas are out to get you! Wish to bring back a dead loved one? You've disturbed the fabric of the universe and there is a price to be paid to some god for this live re-given!
When players ask to be monsters: and they have, I generally limit them to a CR equal to their level or the spell level, whichever is higher. 9th level covers the majority of monsters that won't shatter the game, and allows a few types of dragons, vampires, driders, werewolves and other fun stuff.
Personally if the player didn't jump on asking for something right away, I'd discuss with them outside of the game what they'd like to use their wish for. Compromise is always the best route.