It depends.
Some champions have abilities that "stack," or gain power by overlapping one another or having buffs that persist long enough to overlap. Champions like Sion, Nasus, and Veigar all have abilities that "stack." Every time they kill a minion or champion in a certain way (with the exception of Sion) they become more powerful. Sion gets points added to his maximum health, Nasus makes his Siphoning Strike more powerful, and Veigar gains ability power.
Other things like Twitch's deadly venom can be applied to the same target multiple times to increase its effectiveness. Vel'koz has his passives that activates when it stacks 3 times, dealing true damage based on Vel'koz's level, Braum's passive stacks up and stuns you when you get 4 stacks....
It's not all that common of an ability, but this is the basics.