What's the level cap in Skyrim?

According to Todd Howard, game director at Bethesda, there is a mathematical cap somewhere in the 70s. Although most players will max out at level 50.

See this article for details.

Since you gain perk points each time you level up, and levels only by increasing your skills, the maximum level is when all of your skills are lvl 100.

For my cat I tested this with (using console commands) I managed to reach lvl 81 (and a bit), for a total of 80 perks.

As of Patch 1.9, the level cap of 81 has been removed.

^{(This is an update to the current answers, regarding Patch 1.9's changes.)}
At higher levels, leveling up happens much more slowly. Level increases in Skyrim follow a formula (detailed here). Prior to Patch 1.9, the maximum level was 81, since there was no way to gain any more experience once every skill reached its maximum value of 100.

With Patch 1.9, individual skills can be made "Legendary", denoting them with an Imperial symbol. This will reset the skill to 15, and perks used for that skill may be redistributed. Gaining levels in that skill will affect leveling again, thus effectively removing the level cap of 81. There is no restriction on the number of times that skills can be made Legendary. This change makes it theoretically possible to obtain every single perk in the game.

According to the formulae, training one individual skill all the way from 15 to 100 (one mastering cycle) yields 4,930 XP. This amount of XP is enough to level from 1 to 17, or from 194 to 195. Leveling up to 252 (unlocking all perks) would take a total of 809,475 XP. This corresponds to over 164 skill mastering cycles, or mastering every skill over 9 times. The proper approach to such a huge task would be to find the easiest skill to raise and to exclusively focus on that for leveling purposes.

^{Source: UESP wiki, "Skyrim - Leveling" article}

I'd like to expand on what GalacticNinja stated, an example of an easy way to level up, though this isn't the only way. At the start of The Elder Scrolls: Skyrim, if the player chooses to follow the Imperial Guard (I think) and you are searching for some armour and a sword, it is set so once you open the chest, it continues to the next room. However, were you to instead pick up the sword on the wall and sneak behind the person you followed, you can hit them with the sword without being detected or instigating aggression. It takes roughly 50 minutes to max out the sneak skill, as for the one-handed skill, it should be around 32 the first time.

Once both sneak and one-handed are made legendary at least once and then taken back up to level 100, you have the benefit of being practically undetectable in the darkness while having a killer one-handed skill (If you'll excuse the pun). Once you get a dagger, you will be able to one-hit kill just about anything if they didn't scale up in level to you.

As others have said, you can reach level 81 without making any of your skills legendary, and at level 252, you should be able to unlock all available perks.

However, the hard level cap is 65, 535.

I was curious, after others stated that there isn't a level cap. In computer logic, number values have limits; having no level cap would be next to impossible. I set up Skyrim on my computer, and started playing around with my character's level, using the console.

I was able to observe the following behaviours:

1. When using player.setlevel to set my level to some arbitrarily large number (usually to the effect of '9999999999999'), my level would always default to 65, 535.
2. When using player.advlevel to advance my level past 65, 535, my level would return to 0.
3. When using player.setlevel to set my level to values slightly over 65, 635, my level would roll over. For example, player.setlevel 65636 set my level to 0, and player.setlevel 65545 set my level to 9.

This makes sense, as 65, 535 is the maximum value you can store in a 16-bit unsigned integer (the type of 'container' one might use to store and represent a player characters level in video game logic). When you tell a computer to 'add one' to a 'maximum value', it resets back to 0. This behaviour can be observed a lot more easily in early computer games, as they would often use smaller containers to save memory.

Out of curiosity, I thought I would see what happens if I manually level up from level 65, 535. I used the console to give myself the ingredients required to make 10, 000 invisibility potions. I found the nearest alchemy table, and spent the next 30 minutes spamming R. I made the following observations:

1. Crafting 10,000 invisibility potions allowed me to reach level 97 Alchemy with 6 legendary ranks.
2. Every couple of legendary ranks, the level progress bar would increase by approximately 0.75%.
3. My hand hurts.

Extrapolating this information outwards, I predict needing to hit up to legendary rank 400 before seeing the progress required for the next level. By mass-producing invisibility potions, I predict that this would take upwards of 28 solid hours.