As was noted with one-factor-at-a-time you freeze all of the variables except one at some level and then run experiments at 5 levels for the one variable you are changing. A single iteration of this kind of design is 25 but to get the same precision you get with a factorial you would actually have to run 5 times as many as the full factorial – 15,625. Either way this is far too many experiments.
If we pretend you really need 5 levels (it is possible but I’d want to see the proof) then the better bet would be a rotatable composite design – the basic design would be 2**5 +2*5 +1 = 43 experiments. For perfect orthogonality you would have to toss in a number of replicates of the center point only. You could get away with two or three reps on just the center – the final effort wouldn’t be perfectly orthogonal but from a practical standpoint it probably wouldn’t matter so in a pinch you could do the full analysis in around 46 experiments this would give you information on all of your two way interactions, all of your curvilinear terms and all of your main effects.
If that’s still too much you could run a half rep of the full factorial part of the composite so that would result in 16+10 +1 +3 = 30 experiments this would give you the main effects, the curvilinears, and some of the two ways – but these two ways would be confounded with other two ways. …And, if you wanted you could fractionate the full factorial even further – say a quarter rep – which would result in 22 experiments.
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