How can I increase weapon damage as much as possible in Skyrim?

Use a potion of smithy and have the respective perk for that material (under the Smithing Perk Tree), then upgrade it at a grindstone. Some alter blessings and items boost smithing. The higher your smithy, the better items will upgrade. No glitches required.

There are a couple of ways to increase weapon damage.

1. Get a better weapon, unless you are in end game and have some of the best weapons and materials available to, the next few options might not be worth it

2. Upgrade your weapon on the grindstone at a forge, your going to need some materials to do this, and the higher level you are in smithing, the greater the increase in damage.(Upgrade Ranks: Fine, Superior, Exquisite, Flawless, Epic, Legendary)

3. Put damage enchantments on it, this can increase the damage done by the weapon instantly by 10 damage depending on skill and spell, at level 100 enchanting you can put 2 damage enchantments on it, and with a grand soul gem, get even more damage out of it

4. Put poison potions on your weapons, these don't last long, but it will still increase your damage

5. Like one handed weapons, two handed, archery? Go to the specific skill tree of your favorite and put a few skill points there. The first point in each tree is a 20% increase in damage.

P.S. To get absolute MAX DAMAGE you're going to need enchanted equipment with fortify smithing on it and a potion of fortify smithing, as well as smithing at level 100 so you can max the damage increase on the grindstone

If you want more information on Smithing in Skyrim visit: http://elderscrolls.wikia.com/wiki/Smithing_(Skyrim)

NOTE: In the early days of Skyrim there were a couple of damage exploits, but they were promptly patched, to my knowledge there aren't any damage exploits in the game today

Get a full set of armor with fortify smiting. Then drink a potion of fortify smiting. After that build the strongest weapon and armor you can. hen upgrade them. When enchanting,Invest in enchanting perks, have a full armor set of fortify enchanting, and drink a potion of fortify enchanting. Then enchant the gear with the enchantment of your choice. After that invest in perks for either light armor, heavy armor and either one handed, archery, or two handed. There you go. For the best weapon do this with a stahlrim weapon, and when you enchant, use ultimate chaos damage and another frost enchantment.

I don't think this method is significant after Oak's answer, however...

Eating vegetable soup gives you a steady supply of stamina. Since you only need 1 stamina to power attack, vegetable soup allows you to do then non-stop.

This is the old restoration glitch . This method works the same way with boosting smithing,enchanting and alchemy . You can make any playstyle overpowered this way . It's time consuming but you use less perk points this way . You do have to be careful using this glitch though , you can lvl your character way to fast . Also a great way to make gold for training skills at higher levels .