You may have noticed that there is no Potion of Fortify Alchemy or apparel of Fortify Enchanting. That means you need cross-pollination.
The gist of the loop is very simple.
Equip apparel of Fortify Alchemy to brew better Potions of Fortify Enchanting.
Drink Potions of Fortify Enchanting to enchant better apparel of Fortify Alchemy.
Repeat until you see no improvement.
The loop provides diminishing returns. After a few loops, you won't be able to improve until you've increased your skills.
Getting there!
When you are done looping:
Enchant apparel of Fortify Archery.
Enchant apparel of Fortify Smithing.
Brew a Potion of Fortify Smithing.
Improve your bow and armor. See Ultimate Gear.
Enchant your bow with (Fiery) Soul Trap. The strength of the enchantment is fairly inconsequential as it does not affect damage.
It's over ONE-THOUSAAAND!!
(The game won't even display that many digits.)How high you can go depends on your Skyrim edition and mods. In unmodded Skyrim Special Edition (without the Unofficial Skyrim Patch) the best you can do is:
Apparel of Fortify Alchemy +35%.
Apparel of Fortify Archery +57%.
Apparel of Fortify Smithing +35%.
Potion of Fortify Enchanting +40%.
Potion of Fortify Smithing +158%.
These percentages compound to make a bow that (with sufficient Archery skill) deals over 1000 damage.
(On older versions of Skyrim, the limit is slightly higher or entirely absent. Go as high as you like--or until the game crashes.)
Ahzidal's Genius
(Wear four of Ahzidal's Relics.)You can do the loop before you have all the stuff necessary to reach the end. Do the best you can, see how far you get.
It's tricky, but you have to start somewhere. At around level 20, you should have the skills and means to make it worthwhile. You can buy some apparel of Fortify Alchemy to get a head start, since at this stage the store-bought apparel is still better than what you can produce.
It'll be very slow going at first, but eventually the process starts feeding itself as you'll be making tons of money with which to buy more materials.
Consider doing a loop when:
You have four Grand souls.
You find (or craft) a better bow.
You get the Smithing skill perk Arcane Enchanter.
You get the Black Book: The Sallow Regent.
You get Ahzidal's Genius.
Complete the loops when you have Alchemy and Enchanting skills of 100.
Alchemy 41 Circlet
You'll need a set of unenchanted apparel for the following slots:
Head: Hats weigh the least, but a Leather Helmet will do.
Necklace
Hands: Gloves weigh the least, but Leather Bracers will do.
Ring
You'll need two Potions of Fortify Enchanting. If you work quickly, you can enchant two items with one potion.
To keep track of items in your inventory, name them Alchemy <rating> <item>. For instance, Alchemy 35 Ring.
See Enchanting to make the most of this.
Seeker of Shadows
Mix two of these together:
Blue Butterfly Wing
Hagraven Claw
Snowberries
Spawn Ash
Spriggan Sap
See Alchemy to make the most of this.
Archery 67 Necklace
You'll need a set of unenchanted apparel for the following slots:
Necklace
Head: Your best helmet.
Hands: Your best gauntlets.
Ring
You'll need two Potions of Fortify Enchanting. If you work quickly, you can enchant two items with one potion.
Give your items cool names.
See Enchanting to make the most of this.
Smithing 41 Robes
You'll need a set of unenchanted apparel for the following slots:
Necklace
Torso: Clothes weigh the least, but Leather Armor will do.
Hands: Gloves weigh the least, but Leather Bracers will do.
Ring
You'll also need two Potions of Fortify Enchanting. Before you drink a potion, save and do a dry run! The potion will wear off while you stare at the menu. If you work quickly, you can enchant two items with one potion.
To keep track of items in your inventory, name them Smithing <rating> <item>. For instance, Smithing 35 Ring.
See Enchanting to make the most of this.
Potion of Fortify Smithing
Mix two of these together:
Blisterwort
Glowing Mushroom
Sabre Cat Tooth
Spriggan Sap
See Alchemy to make the most of this.