How many Minecraft mods can you run at the same time?

You can add as many mods as you want, depending on the strength of your PC, and mods compatibility with each other. Don't listen to the other guy, you don't need a "unlimited mods" mod to get as many as you want, he probably liked a virus.

You can run literally hundreds of mods at once… But some mods may interfere with other mods so watch out for that, but you can go download Feed The Beast which is really fun and has a lot of cool mods or you can go download Tekkit. Tekkit has a bunch of mods built into the launcher itself and you can even go to the website and download mod packs you want. Now if you're more of a build in minecraft and like to play in creative mode I'd highly recommend downloading a Bukkit server and getting the plugin "world edit". It's really awesome you can span giant spheres, cylinders, walls you name it you can spawn it in. But honestly you should just download all three of them because they are all really worth it.