If you were to climb to the top of mt. everest (~9km above sea level), how many breaths of air would you need to take at that altitude to get the same amount of air in your lungs as you could when breathing at sea level?

Three (3) times breathe of air

Explanation:

On the earth surface, the tropopause is the interface between the troposphere(the lowest region of the atmosphere, extending from the earth's surface) and the stratosphere(the second layer of the earth surface above the troposphere).

The tropopause lies on average, at 17 kilometers above equatorial regions, and about 9 kilometers over the polar regions

As you climb to the top of the Mt. Everest(troposphere), the oxygen level begins to drops.

Oxygen levels at the peak of the Mt. Everest will be 33% of the oxygen available at the sea level because the distance is 9km above the sea level. It means that one have to breathe three times than a single breathe at the peak.