Increasing wire gauge by twisting pairs
If you twist two wires together, each would carry half the current, so you'd "effectively increase the gauge." American Wire Gauges go down by about 10 for every factor of ten in cross-sectional area. If you had ten #20 wires connected in parallel, they could carry as much power as one #10 wire. With two #20 wires, you'd have the equivalent of one #17 wire.

(A handy "rule of thumb" value: #40 copper wire has about an Ohm of resistance for each foot. By the rule above, #30 would have an Ohm for every ten feet, and #20 an Ohm for every 100 feet.)

Note that connecting wires in parallel may work at DC or low frequency AC. For audio, RF, or other purposes, you'd just mess up the wire characteristics, and cause yourself problems.