What are the benefits of alchemy?

Depends on your play style, But I find there's a few potions that are well worth making.

• Healing Potions: In certain fights I find myself chain chug healing potions.
• Blacksmithing/Enchanting Temporary buffs
• Improved Carry Weight: For when you get to the end of a dungeon and need 10 more carry weight to fast travel back to a town!

As it happens, money isn't a limiting factor, Once you get a few points in alchemy you can level it very easily, and if you do it right you can actually make a profit doing so.

In addition to the other answers given, alchemy provides the single most game-breakingly overpowered exploit in Skyrim. It works because the effectiveness of any "Fortify X" enchantment on your gear is based on your restoration skill.

By enchanting a set of Fortify Alchemy gear you'll get stronger potions. If you use that gear to create a Fortify Restoration potion, drink it, and remove/re-equip your gear, the game will calculate a new higher bonus based on your improved restoration skill. You can then make another Fortify Restoration potion with the higher bonuses and rinse and repeat until your alchemy bonus reaches ridiculous heights. At this point you can create super powered potions of Fortify Smithing to improve your gear, Fortify Enchanting to enchant it, and any other overpowered potions you may desire. With this exploit you can level alchemy to 100 in under five minutes (assuming you have enough ingredients), and make potions worth 10,000+ gold each.

Warning: This exploit doesn't have an upper limit like stacking fortify enchant potions and fortify alchemy gear does. If you make your potions too strong the bonuses will grow larger than the variables the game uses to hold them. This does very bad things and will most likely force you to reload your game. I'd suggest not creating any potions with Fortify Restoration stronger than +100,000% or so.

UPDATE: The unofficial patch fixes this exploit.