I was wondering about this question. What if the only design requirement is having the highest steepness of the roll-off but using only one op-amp? I know it's possible to construct approximations of second order low-pass filters with one but what about higher orders?
You can make a high-order low-pass filter with no op-amps at all. In principle you could obtain any roll-off rate you like, given enough pi or T sections. You could then add an op-amp output buffer to make your filter a "one op-amp" filter if for some reason you'd like to do that.
But high-order filters tend to be very sensitive to small changes in component values, so they often require hand-tuning each circuit to achieve good performance, and the performance could drift as temperature changes.
The benefit of using active filters is that the op-amp allows low output impedance and high input impedance, reducing the loading effect of one stage on the next. This means you can cascade several first or second order filters without much interaction from stage to stage, and reduce the need for tuning. If you try to make a high-order active filter in one stage, you're giving up a key advantage of using active filters.
You can make a 10th order filter using an 8 pin SOIC chip like the LTC1569-6
This doesn't use any conventional op-amps but, keeping in with the spirit of the question and @thephoton's answer you might regard this chip as counting as at least one op-amp (equivalent).