Saturating BJT on different value than 10
Short answer:
Yes ...saturation can be reached at higher \$HFE_{forced}\$ than ten.
However, it is risky. Risky because HFE varies:

from transistor-to-transistor
with temperature
with current magnitude flowing through collector-emitter.


Current gain in the table above is specified @ 25 degrees C. Note that at high currents, the measured gain is done with a pulse test to avoid heating the chip.
A designer might survey this table and see that worst-case current gain is only 35. It is even lower if saturation current is above 500mA, especially if the transistor runs hot (which is very likely).
The data sheet graphs (below) showing saturation region are taken with a &quot;nominal&quot; transistor, whose HFE is quite high, certainly much higher than worst-case. Perhaps 150-200.
Should you happen to apply these graphs to a transistor whose HFE is near minimum (like 35), then saturation can very easily be missed, the consequences being failure due to overheating. The circuit function may fail due to collector-to-emitter voltage being too large.

from ON semiconductor data sheet: http://users.ece.utexas.edu/~valvano/Datasheets/PN2222-D.pdf
 So if you happen to select a nicely-high HFE PN2222A transistor (like HFE=200) then you only need to supply 0.1mA base current to saturate it when driving a 10mA collector or emitter load current. For such a transistor, the rule-of-thumb overdrive factor-of-10 is overkill.
But woe be to you if HFE is lower, or temperature drops to -55C.