How high do the levels go? I'm at level 20 with no end in sight...

I don't think there is a cap on how high your level can be, but the highest level required for baking certain items is currently level 17.

Knead to Bake Guide to Fantine&;s Bakery

I don't suppose there's necessarily a reason to put a cap on, either, since that leaves open the option to always add higher baking stuff or achievements or whatever.

You can get quite high a level in the Blue Building too, but there it doesn't help you at all. The most you need to see all the items there is 20.

It's been claimed, and I have not observed so the effect is small or I am so very unlucky, that being higher level makes you less likely to burn lower level stuff. So one reason a user might choose to get a higher level of baking would be to make it easier to bake lower levelled foods.