Maybe not saturate in the way ferrite saturates. But the current can't keep rising indefinitely - eventually you start getting resistive losses and the inductor turns into a (low value) resistor. Then it gets hot, catches fire, and turns into a very high value resistor...
For a given number of turns, an air core will have a lot lower inductance than a ferrite core. So if you want it small, it will have to run at a higher frequency. Or you could make it big with thick wires, like the air core inductors used for speaker crossovers.
The exact brand of inductor would not be critical. It just needs the right value in micro henries, and thick enough wire that it can handle the amps. The DC resistance should be as low as possible. Searching through parts catalogues, they usually list the max current. 3 amps is going to be a chunky sort of inductor maybe an inch cubed with thicker than 1mm wire.