Cockroaches of course. "Tenacious" makes me think of squid, octopi, and barnacles. In this context, I think you mean "insidious." Poison them, and they will die, but more will be back even before you have forgotten about them. Leeches can be quite tenacious.
Cockroaches and rats (wild) can be very difficult to get rid of as well, plus the cockroach can survive an atomic bomb. Fortunately I have never experienced any of those infestations, though we are currently dealing with an invasion of small black ants- even after having an exterminator treat our house.
The cockroach is the most tenacious creature on earth. It has been on earth since the time of the dinosaurs. It can eat anything and can nibble at plastic too. Also it can survive in very extreme climates and go without food for very long times.