In a custom campaign setting a few years ago, one player wanted to play a tiefling...really, really, really wanted to play one. But tieflings didn't exist in my game world, and I was very clear about that from the beginning. There was only one other plane, the Feywild...there were no such things as angels, demons, or devils, and there never was. (I suspect that was why the player wanted to push so hard for tieflings in the first place...he saw they weren't allowed, and that made him want it. Some players live to frustrate the DM.)
So as a compromise, I allowed him to play a wood elf that had all of the abilities of a tiefling instead of elf. Basically, he got to play a tiefling in every way except flavor: we would never use the word "tiefling," there wouldn't be any references to demonic/devilish heritage, hellish rebuke was renamed to something like fiery rebuke, etc. He could even look like a goat-person with hooves and horns and whatever if he wanted, but it would be more Feylike than fiendish. Think satyr or Pan, instead of imps and Asmodeus.
He agreed to the changes, only to abandon it a half-hour later and roll up a dwarf instead. --sigh-- Ah well.