Types? Heck, I'd say an easy 20 main categories, and then based on very specific ingredients, brewing method, even geographical location, and all the "hybrid" and "in between" options, triple that just to start 🙂
Beers themselves, as in, a breweries product, wow, a bunch. Heck, in my little corner of the world, there's 5 breweries within 7 miles, and ~20 if you open up the distance to just 30 miles. Each with typically 6-8 in house brewed at any given time. I bet the liquor store I go to for the big beer selection, a walk in cooled beer cave plus ~10 wall coolers, all with beer, probably has 250-300 different distinct beers.