Let me preface this by saying that I'm too often pennywise and pound-foolish, and yes, that means I stoop to pick up pennies off the ground more than most people will.
So what do I do with all those pennies? Sometimes I use them to pay exact change at the store; sometimes I give them to beggars in a handful; sometimes they pile up to such great quantities that I can't bear to have them jingle around in my bag or pocket all day, so I empty the mug brimming with pennies, nickels and dimes (quarters are strictly for the laundry), and walk the comparatively short distance to the neighborhood post office.
The vending machine that spits out stamps at many U.S. post offices is one of the few that will actually accept pennies (which is to be expected from the government that gave you pennies in the first place). So I save myself the rolling of eyes I'm likely to get from real, live cashiers and instead pump my pennies and other small change into the silent and accepting vending machine to buy stamps. I know this isn't going to save me vast quantities of money, but it's satisfying on some level to get rid of the change and come out with thin, little currency of a different kind.