Practically speaking, I select the highest gauge (regular) insulated wire I have in stock... buying spools of wire gauge higher than needed for that, doesn't seem as cost effective to me as buying more of a gauge that can handle *most* low current projects, not having to (invest in) stock wire that's barely capable.
However I question the use of 5VDC, presuming that a simple hobby circuit would use a resistive current limiter for each, so you're losing [ (5V - 3.4V) / 5V ] * 100 = 32% power as heat at the resistors alone. If you were to switch to using a 12V supply and ran 3 x LEDs in series for 10.2V sum forward voltage, you more than cut your resistor loss in half. j Granted, 3 x series doesn't divide into 10 evenly, but you're running them at under 1W each so perhaps running 9 LEDs at 222mA each instead of 10 LEDs?