polling.py

class APIWrapper:
    
    #snippet...
    
    def poll_api(self, tries, initial_delay, delay, backoff, success_list, apifunction, *args):
        time.sleep(initial_delay)
        for n in range(tries):
            try:
                status = self.get_status()
                if status not in success_list:
                    polling_time = time.strftime("%a, %d %b %Y %H:%M:%S", time.localtime())
                    print("{0}. Sleeping for {1} seconds.".format(polling_time, delay))
                    time.sleep(delay)
                    delay *= backoff
                else:
                    return apifunction(*args)
            except socket.error as e:
                print("Connection dropped with error code {0}".format(e.errno))
        raise ExceededRetries("Failed to poll {0} within {1} tries.".format(apifunction, tries))

You can just use delay directly, no need to "make it mutable". In fact, you can never make an int (or float) mutable, but you can assign a completely new object to a variable.
The line delay *= backoff would create a new immutable number object with value delay * backoff, and then assign that to the name delay. The old value of delay is forgotten.

wouldn't using delay directly cause the next call to poll_api use the old value of delay then?

Oh wait, I think I just answered my own question. No it wouldn't, because delay is immutable.

However, for parameters that are mutable, this could cause a problem I think.

Code updated! Thanks encukou!