No limit, any tyre sized 26" will fit although there are always minor variations, meaning that some tyres are easy to fit and others can be a nightmare. New tyres are always squeaky clean and sticky so when you get your new tyres, sprinkle the beads with talc, which is the best dry lubricant for rubber. This will help you fit them.
Go for a slick or an all-terrain tyre with a narrow central ridge in about 1.6 size. If you go for something smooth and pump it up to the maximum pressure shown on the sidewall you'll be amazed at the difference in speed and probably quite annoyed at all the energy you've been wasting pushing those fat knobblies! All that buzzing noise is the sound of those knobbles squashing around and wasting your energy. Even if you didn't fancy slick and you went from a 2.35" knobbly to a 1.8" knobbly like a Panaracer Fire XC Pro you'd find a big difference and in some conditions a narrow tyre is better off road because the higher ground pressure means it digs down into a soft surface to find grip.
If you're getting serious about cycling, a track pump is a fantastic investment as it has a gauge and enables you to achieve much higher pressures than a simple hand pump. Have a look a the Topeak Joe Blow, these are very good. Don't make the mistake of buying an MTB-specific track pump because the barrel will be fatter and if you ever go over to road riding you'll find it very difficult to pump up a road tyre to 100-120 lbs
