There is frighteningly little strict API documentation (read: ZERO) for multiprocessing.pool.ApplyResult. The multiprocessing explanation doc talks ABOUT ApplyResults, but does not define them.
The same appears to apply to multiprocessing.pool.Pool, although the Python multiprocessing guide appears to cover it better.
Even the ApplyResult help() results are paltry:
| get(self, timeout=None)
| wait(self, timeout=None)
Your right that there is in a glitch in the documentation: the class is actually documented as AsyncResult, not ApplyResult. The two are different names for the same class:
>>> multiprocessing.pool.ApplyResult is multiprocessing.pool.AsyncResult True
The name may have been changed at some point and the docs weren't consistently updated, but everything is documented, it's just documented under the wrong name. (There is a closed bug in which someone pointed out that the docs mention AsyncResult but the class is actually called ApplyResult, so they added AsyncResult as an alias.)