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.
I found some information about Alchemy on the Elder Scrolls Wiki
According to the Wiki, Alchemy has been confirmed as a skill in Skyrim. Alchemy will govern the ability of the player to make more potent potions and poisons.
Instead of carrying around the tools (retort, alembic, etc) players will instead utilize labs or alchemy stations throughout the game.
Each possible ingredient in Skyrim has been confirmed to have four possible properties. These are unknown until the player either individually eats an ingredient or successfully mixes a potion/poison.
Alchemy ingrediants can be found throughout Skyrim growing in the wild as well as in towns. They may also be found through certain shopkeepers.
Best Answer
It works like this: Certain things in the game world scale with your character linearly, like they did in Oblivion, and certain things are "semi-static" and "semi-scaling", meaning that they scale to a certain extent, but the scaling has limits.
The latter one consists mostly of dungeons. All of the various dungeons in the World have a static, pre-determined difficulty which is described in terms of a range of the players level.
For example, there are a lot of dungeons early on which have the difficulty level "1-10", and that's where they'll always be, they won't scale beyond that, and similarly, dungeons of higher difficulty level will not be manageable by you until you level up some more.
This is also the reason why the game still feels very leveled despite having a different scaling system. A lot of the dungeons and areas that you encounter around the part where you start (southwest near the mountains) and north towards and around Whiterun are in this level 1-10 area, only slowly progressing more upwards around the edges of that area. For example, the first dungeon with a higher difficulty which can be reached with ease is, for example, a dwemer (practically the "tinkering dwarfen" race of the Elder Scrolls games) ruin that is accessed via two levels of caves just to the north-east of Whiterun, embedded into a small mountain. And once you start visiting the other places, you will usually have leveled beyond level 10 and the game world will scale fairly well with you.
An exception to that is the case of "cleared" dungeons and possibly areas. When you have previously cleared a dungeon, it will say so on the map when you hover the cursor over the location's legend. These cleared dungeons will eventually get repopulated after a long time (it's longer than just a couple of days, but I don't know the exact number), and when they do, they repopulate with mobs in a level-range that corresponds to the players level.
The actual level of the monsters inside those dungeons is determined by your first entry into the dungeon. When you first enter the dungeon, the level of all monsters inside it is "locked" to your level. For example, if you enter a dungeon of difficulty "10-20" with level 14, all enemies will be set to level 14 and will be locked there, so that if you find it too hard when you first enter it, you can come back later and try again with level 18, at which point they will still be level 14.
The difficulties in that case are clamped, which means that if you enter a 10-20 dungeon with level 5, all enemies will be set to level 10, and if you enter it with level 25, all enemies will be set to 20.
Also, the areas in Skyrim work roughly that way, too. In general, you will find more difficult monsters the higher you get in the world (near the mountains). An obvious example that comes to mind are the frost trolls and sabre tooth cats, which will beat you up badly, if you walk around in the mountains too much early on. But as opposed to dungeons, monsters won't be locked into a level once you enter the area. Instead, the game randomly sometimes spawns monsters that are a couple of levels below or above you, or equal to your level.
One of the things which has "linear scaling", almost exactly like in Oblivion, should be dragons. Todd Howard has said a couple of times that he wants every player (no matter how good or bad) to be able to experience at least all of the main storyline. Although even with those, it will probably be an advantage to have better equipment or perks which give you some kind of resistance to something (for example, there are frost and fire-dragons, and a couple of other types).
An example of monsters who are probably completely static in level are giants and mammoths. Ever fought a giant with Level 1?
Notice that this only concerns the level scaling of monsters and other enemies around the world of Skyrim. As pointed out by SLC in the comments, items seem to fully disregard the semi-static level scaling of dungeons and simply scale linearly with you:
In addition, there are scaled items, which are usually quest rewards and depend on the players level at the time of quest completion, similar to how dungeons scale.
Also, this might count as kind of a second-hand source, but at least it's an interesting read. It's a little explanation of how the scaling worked in Fallout 3. If you Google for "Skyrim scaling" you will find a lot of references to devs saying that the level scaling is somewhat similar to that of Fallout 3.
Enemy level generation, from UESP wiki, "Effects of Leveling":
Also, here's a discussion on the same topic which might be interesting to read.