There's no answer to that. Every facility is different. A Center controller joined the discussion and said he'd worked 59 planes at one point (!!!). Here at P31, we typically wouldn't have more than 15 airplanes on a scope at once, though I've seen higher numbers than that on occasion.
Here's what I added to the thread:
It's not necessarily the number of planes, but what each of them is doing that makes things complicated.
If you have 12 airliners all wanting to land full-stop at the same airport, it's fairly straightforward. It's sequencing work, of course, but nothing strange. They all just want to land, and you just need to line them up somehow.
Now if you have...
- Three military trainers doing multiple practice approaches and constantly changing their minds. Two of the students are foreign nationals, whose English is barely understandable and who disconcertingly answer every instruction - from approach clearances to traffic calls - with only a "roger".
- One fast-moving F-15 with an automation issue who won't handoff to anybody, ten miles from the boundary and screaming along at 450 knots.
- Three airliners trying to get down from your northeast gate, weeding through a swarm of VFR targets maneuvering in their way. It's like Resolution Advisory Bingo. Then you need to sequence them with your military practice approach trainers.
- Two other airliners trying to climb out through the same northeast gate (playing chicken with 737s is fun!) and through the same VFR targets.
- A military trainer doing aerobatic airwork right in the middle of your southern arrival gate.
- A Lifeguard priority Lear Jet trying to beat the airliner pack in while dodging the aerobatic guy.
- An ornery Mooney pilot who's bitching about getting vectored around so he doesn't get run over by the jets.
Those are just the guys you're talking to. Add on to that point-outs from other sectors - like military jet trainers climbing high through your airspace out that same northeast gate or a pair of slow-as-molasses IFR helicopters cutting across your final, trying to get to another sector - and it gets more fun.
That pretty much describes an average session on our East sector.