The delight of flying out of an airfield with full ATC is in exactly this situation. Told we're number 3, then as we come abeam the threshold on the downwind "Can you accept a tight circuit?"
"Affirmative".
"Cleared to final number one, turn in now."
Happens quite often, and usually we're down, off the runway, down the taxiway and onto the apron before the next one comes over the fence.
I watched a Twotter do a tight approach a few years ago - I don't think he went outside the airport perimeter, turned final at something like 250 feet and half a mile, landed, and off at the first intersection. Pure poetry in motion. Almost slow motion.