I heard somewhere that King Kong would be much too large to climb at all. Elephants and hippopotami are extremely poor climbers. I know that éléphants are good swimmers but the same cannot be said to hippos.
How about the very real Gigantopithecus blacki?
This Ape could indeed climb if it desired to. Presuming suitable hand holds of course.
Even assuming a larger Ape, Ape's are evolved from climbing creatures. The only reason why they do not climb more, is that most tree branches are not strong enough to carry their weight.
In terms of trees, the upper-bound seems to be Orangutans (or Bosi lemur - if we are considering primates in general), which are not small, but are still restricted to the sturdier branches of the tree.
Elephants and Hippos are understandably poor climbers, they never evolved for that purpose.
Both of their ancestor though were aquatic, having sister families of highly adapted aquatic mammals (Dugongs/Mantaees for Elephants, Whales/Dolphins for Hippos). It also makes sense (evolutionary) that elephants are the better swimmers as the elephant's ancestors were further along the aquatic path than the hippo's ancestor when they diverged, and hippos have not had the need to develop further aquatically.