I'll provide a detailed answer about dice usage in D&D 3.5 specifically, matching the question's tag.
In D&D 3.5, rather large numbers of dice can be rolled at once. The core spell Disintegrate, for instance, deals "2d6 points of damage per caster level (to a maximum of 40d6)". This type of spell is generally considered a "save-or-die" scenario, where the exact total of damage is unlikely to matter; but it does exist, and could come up.
Now, for some specific trends related to dice:
The dice types used in D&D are d2, d3, d4, d6, d8, d10, d12, d20, and d100 (often written as "d%"). d1000 is also occasionally used in some supplements, but I don't recall any use in the core rules. I'll note that the d2, d3, and d100 are not physically present in most physical dice sets (in a non-digital setting, they're usually emulated using other dice), but are used throughout the rules nonetheless.
In terms of large quantities of a single kind of die (possibly mixed with others), I'd say that d6s are the most common, but d4, d10, and d8 are all seen as well. Spellcasters and the Rogue's Sneak Attack ability regularly use large numbers of d6s for a single damage type, scaling at a rate of 1d6 per 1 or 2 levels. The spell Magic Missile is a case where many players will end up rolling 5d4+5 at once, although strictly speaking that's a case of 5 separate 1d4+1 rolls.
You'll also encounter large quantities of mixed dice, including cases where it's important to distinguish different sets of the same type of die. This is more common with martial fighters, and again also some rogues - anyone who uses weapons, really, due to the various magical properties that add extra damage dice. For instance, a critical hit from a +1 Shocking Burst Thundering Dragonbane longbow, as wielded by a 14th level human rogue, would involve rolling dice for
3d8 piercing + 1d6 electric + 2d10 electric + 2d8 sonic + 2d6 + 7d6 damage (plus a constant amount added on top).
If you want to support epic-level characters (which is a dramatic uptick in power), then be aware that there are published spells dealing 305d6 damage on a failed save.
Other answers have touched on the programming aspect, and I agree that "many dice in assorted groups" is the preferred standard for a general-purpose dice roller. How much is "many"? Fairly large, it seems.