Maxing Out Perks

Since patch 1.9 there is no maximum level anymore.

If you have a skill at 100 you can make it "Legendary". This resets the skill to 15, gives all perk points invested in that specific tree back. Be warned: You lose all the benefits of having a higher skill level! The advantages of making skills legendary are:

• Difficulty increase, since the enemies don't have their skills reduced.
• You can continue to level up and earn extra skill perks.

It is possible to repeatedly make a skill legendary or to make multiple skills legendary, this allows you to (slowly) unlock all perk points.

Level 81 is the maximum level in Skyrim without making any skill Legendary

There is a linearly constant amount of "experience" required for each level. The formula is simply:

(level-1) * 25 + 100

Where level is your current level.

"Experience" in this case is earned strictly through skill ups. Leveling a skill to X will give you X experience towards your next level. For instance, to level from level 1 to level 2, you require 100 experience. This means you can either raise 5 skills to level 20, one skill to level 21 (assuming it starts at 15), or any other permutation thereof.

Also, and this is important:

There is no soft cap

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Level 81 is just as (relatively) easy to reach as Level 11, it just requires more total experience.

Since all races have +10 to one skill and +5 to five skills, this means there is a total of 88,085 "experience" to earn, and this is constant across the ten races. Level 82 would require 89,100 total experience, and is currently unreachable.

Thus, when your character maxes all of their skills, they will be level 81.

You can only train skills 5 times. e.g. train sneak 5 OR train sneak 3 and pick-pocketing 2. After that, you must gain a character level. Upon gaining another character level, you can train skills 5 times. These 5 trainings can do not have to be the same skill.