Phrases like, "as fast as humanly possible" are pretty well understood, but what about when you're not talking about a human? A dog can run faster than a human, for instance. How would you express that a dog arrives as quickly as can possibly be expected of a dog? Perhaps a rabbit can't get as close as a human can to say, a high shelf. Is there a way to fill the blank, in that example, for "as close as _________ possible"?
I would use "physically" here, rather than attempting to create new words from each individual.
To me, 'the dog ran as fast as physically possible' sounds better than 'the dog ran as fast as doggedly possible' given the fact that 'doggedly' is already used as an adjective/synonym to tenacious. This can also be applied to other animals that do not lend themselves well to such structure, eg "The rabbit reached as high as physically possible" sounds much nicer than "The rabbit reached as high as rabbitly possible."
You could also use "naturally" in this sense, eg, "the rabbit reached as high as naturally possible," meaning as high as it could do without artificial assistance.
One option might be to use adjectives associated with certain types of animals, e.g. canine, feline, etc. You could append -ly to make them adverbs like humanly, like, "It ran as fast as caninely possible." While these aren't quite as natural as humanly in everyday speech, I think you'd be unlikely to confuse people with such terms.
In the context of the question we probably deal with real animals (a running dog), then it is possible to consider: "as fast as nature made it possible".