What's the largest size of "door" I can open with a single switch?

You could make a 5x5 piston door using stick pistons - but if you're new to Redstone this may not be for you.

There is also an option of making a "gravel door" (I am unsure if this works in the latest version of Minecraft).

A slightly easier piston door may be a 3x3 version.

Finally, you could use the design in my question here, the 2 high x 3 wide door, but make the door a lot longer - say 2 x 30 by wangling pistons and repeaters. If you wanted to "release" Endermen you'll need a bigger door, however.

In theory, there is no limit on the size of a door. You could expand practically any redstone design to control a larger door. Just add more pistons.

In practice, you are limited by the number of 16x16x256 block chunks the game loads into memory at one time. Redstone circuitry will not work in unloaded chunks.

In multiplayer mode, a grid with a default radius of 10 (for a total of 21x21 or 441 chunks) is loaded around each player and sent to the player by default, although this radius can be configured to be between 3 and 15, usually only lowered with a poor connection home server. These chunks may have activity (mobs spawning, trees growing, water flowing, dropped items disappearing etc.), while the other world chunks are inactive, stored on your hard drive.

Without going to extreme levels of complexity:

24

A piston can only move a stack of 12 blocks at a time, therefore any mechanism that successively pops blocks onto/off of a stack to open/close a door will have a limit of 12 blocks. Since you can have two doors parting in the middle for your entrance, 24 is a feasible number for a door that isn't too complex.

For completeness, assume that these doors are vertical, and run all the way from level 0 to the world height. This makes your door size:

24 x 256

on an appropriately configured server.