The how:
1. Make a Fortify Restoration potion.
2. Drink the potion.
3. Re-equip your Fortify Alchemy gear.
4. Repeat from step 1 as desired.
The why: So, in short, the game considers apparel enchantments to be Restoration effects. This means if you chug a Fortify Restoration potion and re-equip your gear, you keep the boosted effects. Including Fortify Alchemy gear. You can repeat this cycle of "drink potion, re-equip gear, make new potion" however long you desire, it gets exponentially more powerful the more you do it. Eventually it might be possible to crash your game from the effects getting bigger than it can handle, but for giggles I was wearing a total of +20,000% Fortify Alchemy equipment after a few minutes.
You can, of course, make Fortify Enchanting/Blacksmithing potions while under these effects for needlessly huge boosts there. I was already one or two-shotting everything on Master with "legit" Smithing/Alchemy/Enchanting, this just guarantees a one-shot on absolutely everything when you use a potion of over 9000% Smithing to enhance your weapons.
Since Alchemy and Speechcraft level up based on the value of the items, this will also max these two skills out absurdly fast. I started a new character just to see how fast, and got from 30 to 100 Alchemy in a single potion worth 300k. Giving that potion to a vendor raised my Speechcraft from 15 to 82, a second one brought it to 100.
This will svck any remaining challenge out of the game if you do it, of course. I mean what are you going to do with a 20 minute duration Paralysis poison? I found it funny that it's essentially a repeat of the Morrowind "Fortify Intelligence" potion alchemy exploit, though. They never patched that out of Morrowind, so there's a possibility it will remain here, too.
