As a total noob to cycling the thing that really kills me is hills steep enough I get into anaerobic territory. My general fitness lets me cycle for hours without really getting tired even working fairly hard, but my thighs have a pretty limited total amount of ascent in them which takes a long time to replenish.

I feel like on steeper hills (say 10% and above) I have to go anaerobic to get enough speed... if I shift down to my lowest gear at my regular cadence I am so slow I risk falling off, and of course the climb just takes much much longer anyway!

My beginner question is: should it be possible to get up most/all hills aerobically using gears, or is it a fundamental feature of cycling that you *will* end up burning, and you *will* find sometimes your legs just give out mid-hill until you improve your muscles/fitness?