This is a problem which has vexed me for years - but no more! A normal stapler will staple up to ~10 sheets easily enough, but given many more, even as little as 18, it's defeated. This is despite the facts that the staple itself has long enough arms to fit through the bundle, and that the stapler's jaw is big enough for the bundle. The problem seems to be that the staple, when forced into the thick stack, bends, rather than penetrating.
The solution is really quite simple: make holes for the arms through the stack before stapling. For this, you will need a sharp implement of roughly the same diameter as the staple's arms, but much greater rigidity. My preferred tool is a 21-gauge syringe needle; a 25 will do, but a 21 is that bit wider. Yet wider needles may be even better, but would involve a trip to stores. A sewing needle is almost certainly also effective.
Getting the holes the right distance apart is critical; this is easily done by using a staple as a ruler, or, if you've already tried and failed to staple the bundle in the normal manner, simply use the failed staple holes as guides for the needle. The best way to do the puncturing seems to be to place the bundle on top of a soft but reasonably rigid object, like a lump of expanded polystyrene or a seat cushion or something, and then push the needle through the paper into it; this keeps the paper flat, whilst not buggering the needle.
Maybe this is a well known trick; to me, it was a revelation, and one i thought was worth sharing. THERE IS NOW NOTHING WE CANNOT STAPLE.
See also: how to solve sliding block puzzles.