For example, warming the climate tends to melt ice. Ice reflects sunlight back to space better than liquid water or dirt. So the reduced ice cover makes more sunlight get absorbed by the Earth. Absorbed sunlight becomes heat, which raises the temperature. The increased temperature melts more ice.

Technically all of these loops can only result in a sigmoid curve, not an exponential. The temperature won't become arbitrarily large. Trivially it can't get any hotter than the input (the sun). But in the ranges we care about (temperatures suitable for human habitation without massive adaptation and migration needed) it's effectively exponential.