Lesse now...
On a airgun that is normally rated for a single stroke, most of them don't even have an air chamber to store the compressed air in, so pumping it a lot will mostly just make you sweaty. Pressurize the air. release the pressure, repeated. The air the pushes the pellet out, is stored under pressure between the piston face and the valve.
Easy way to tell if yours is like that (aside from reading the ****ing) instructions, is that the pump arm will pop out when you go to pump in another stroke.
On a gun like the Crosman's (wood and steel, not the plastic toys)or Benjamin's, pumping over the recommended amount will slowly reach a point where the air pressure cannot force more air into the storage chamber. A really close fitting and properly adjusted pump cup seal will get you a slight improvement. Some guys have increased the amount of volume of the storage chamber, and this increases the total number of strokes worth of air that it will take, as well as increasing the power. It also makes the rifle need to be pumped a lot more times to just plink with.
Some of the Crosman's have a pressure release valve built into their storage chamber to vent pressure over a set amount. More pumps is just exercise. See comment to follow re: pins.
The linkage pins on the Benji's are a weak point, and if they do not get the lube they need, you will need new ones soon. And a new pump arm. The Crosman ones are a bit better, but still need the lube.
The pump arms can buckle if you overdo it.
DAGS for Steroid Benjamin Billet
Mostly, don't cry when it breaks. If you follow the instructions, it'll usually work as advertised.
Cheers
Trev