How many charms can I carry at once, and if I can fill my inventory, do the charm benefits reach a limit after carrying a certain amount? If not, can I theoretically fill my inventory with Annihilus charms?
The Annihilus charm is a Unique charm, so no, you can't fill your inventory with those, since you can only carry one of each at any time. Other than that, for the other charms there's no limit other than your inventory capacity. That's why charms are considered very valuable items: in a limited space they give you powerful bonuses.
Some links with confirmations of the stacking properties:
Regarding the limits, you get a cap only for skills that have a cap (for example elemental resistances are maxed out at 100%); others, such as Magic find, have no cap, so you can hoard as many MF charms you can, but they have diminishing returns; others instead, as the +skill ones, have no cap at all, so for those the sky's the limit!(or, more realistically, you backpack)