MTGNexus

Can I pay the evolve cost multiple times (while on the stack) to put +1/+1 counters on a creature multiple times, or will they all check separately and only get +1/+1 counters once?

Evolve is a triggered ability, not an activated ability. I assume you mean adapt.

The short answer is you are free to activate an adapt ability as many times as you like, but generally speaking you'll be able to put +1/+1 counters only once. If you control Incubation Druid with no +1/+1 counters and activate its adapt ability, then do so again while the first ability is on the stack, the second instance will first resolve and put three +1/+1 counters on the Druid, and then the first instance will resolve, see that there are already +1/+1 counters on the creature, and do nothing.

The exception here would be if all +1/+1 counters were somehow removed from the Druid after one instance of the ability resolved and before the other did.

Monstrosity works the same way, except removing +1/+1 counters does not make a creature no longer monstrous.