According to the Minecraft Wiki article on enchanting, with full bookshelves, the max level is 50, so all further levels are pointless. While this is pretty straightforward, it's not too efficient to get to level 50 before enchanting, since experience yields diminishing returns.
I am going for maximum enchantments, aiming for an enchanted pick with 2-3 enchantments of high level. What's the lowest level where I can achieve this, and does getting to level 50 actually improve the chances of getting these high-level enchantments?
Best Answer
In Short, yes, getting to level 50 gives you the best chances of multiple high-end enchantments.
In Long:
You're kind of riding a grumpy horse on this one. You want the most efficient level to get enchantments. That would be level 50, there is no ifs, ands, or buts about it. Flat out, a higher level gives you the greatest chances of getting multiple enchants. Period.
You can still get the highest level of specific enchants at some levels lower than 50 depending on the type of item you enchant, but generally speaking, I would strongly suggest you enchant diamond tools\weapons\armor only, because they last longer so you get more bang for your 'buck' of experience. As listed on the wiki, Diamond Items don't have the best qualities for enchanting, but 1,300+ uses from a Pickaxe of Fortune III is just too good to pass up if you ask me.
Multiple Enchantments
The formulae for multiple enchants works like this:
Now what this means for you is that the higher your MODIFIED LEVEL is during your enchantment instance, the greater chance you have for the keep going. And those two words are what gives an item multiple enchantments.
In the second step of the process, you can see that it divides your modified level in half. I will go into what a modified level is later in the post, but let's just say it's your actual level (50), for the sake of argument. Now divide 50 in half for twenty five, add one to it (26) and throw it on top of /50. That gives you just a bit better than a 50\50 shot on two enchants for your weapon. That's not bad. It gets much much worse as you keep going though. Each time the respective chances are halved. Rather than do all the math here on Gaming.SE the wiki states:
You can see there that at level 50 you actually have a Greater chance to get two enchantments than you do to get one. And only 16.10% chance to get three. If you want to get three your best chances are at level 50, and they are still not very good.
Specific Enchantments by Level
This part is somewhat less tricky because the wiki provides us with a wonderfully easy to read graph. I however, suck at Microsoft Paint, and cannot make a sexy.jpg of it. I'll do my best here.
(Modified Enchantment Level in parentheses)
This section is pretty straightforward. If your modified enchanting level falls into any of these number brackets you have the possibility to get that enchantment.
Some notes:
This part is decided before the number of enchantments are decided. So the halving of your MEL (modified enchantment level) doesn't affect this adversely. All possible enchants are decided before the # of enchantments comes into play.
If your MEL falls into TWO different level brackets of the same enchantment, then the highest possible level is used.
You cannot get Silk Touch and Fortune. Getting one automatically disqualifies your item for the other.
Modified Enchantment Level
All that stated, you're going to need to know how to get your MEL. Which is easy.
Enchantment Level = Your Experience Level
Random(0,enchantability) + 1 = a random number between 0 and the items enchantibility, and then add 1 to that.
(enchantibility is an items enchantibility determined by the type of item it is, diamond, gold, wood, etc. These are listed on the wiki as well.)
Once you have your MEL from the above formulae, the game modifies it further one last time, either increasing or reducing it by a random percentage that caps at 25% more or less than it was originally. (100 can turn into anything from 75 to 125)
I think this is everything you would need to know. Let me know if I missed something in comments and I'll be sure to add it in or clarify it.