If you submerge a bottle of water in a boiling pot of water, why doesn't the water inside the bottle start to boil as well?

To figure out why this happens, you need to think about what boiling is, and how it works.

As you would know, the water in the pot boils because its temperature was raised above the boiling point by the flame. This required a net transfer of heat from the flame, through the pot, to the water in the pot. Why did the heat flow in this direction? Because the flame is hotter than the water in the pot, even when the water starts boiling ($T_{flame} > T_{boil}$)

Now, think about the water in the bottle. The only way for it to get heat is through the water in the pot. As long as the temperature of the water in the pot, $T_{pot}$, is less than $T_{boil}$, it is still liquid, and it transfers some heat to the water in the bottle. The water in the pot boils off at $T_{boil}$, and can no longer transfer heat as efficiently to the water in the bottle.

This effectively means that the water in the bottle is restricted to a maximum temperature of slightly less than $T_{boil}$, and that is why it never boils.

Another way to think of this is, there must be a temperature difference for a heat transfer to take place. Since the maximum possible temperature of the pot water is $T_{boil}$, the temperature of the bottle water can never exceed this.

EDIT: Another factor to consider is the low conductivity of glass, which means a high temperature difference is required to let a small heat flux through.