Zachski: In the first game, there is precisely one obstacle that you absolutely cannot overcome without Climbing, and that's getting into the town after the gate is closed.
Paviel: Technically two: The other is getting into the castle at night.
The only reason I didn't count that one is because there's really no reason to do so other than to discover that there's no reason to do so. Can't break into the castle, can't always sleep at the stable (if you haven't mucked the stables that day, which if it ticks over to midnight, counts as a "new day" for that purpose)...
The amusing part, however, is that if you try to sleep in the castle grounds (outside of the stables), no matter what your skill in climbing is, your character will suddenly know how to climb over the wall, for just a moment, hehe.