How thin is our atmosphere relative to the size of Earth?
The answer to your question depends on where one defines the edge of the atmosphere.
The Earth's atmosphere is layered. The thickness, density of composition of each layer varies. The five layers are:

Exosphere: 700 to 10,000 km (440 to 6,200 miles)
Thermosphere: 80 to 700 km (50 to 440 miles)
Mesosphere: 50 to 80 km (31 to 50 miles)
Stratosphere: 12 to 50 km (7 to 31 miles)
Troposphere: 0 to 12 km (0 to 7 miles)

Being the outermost layer, the exosphere is the layer with the most nebulous upper boundary. It is also the layer with the lowest density and it is primarily composed of very low density hydrogen and helium, particularly in the outer/upper reaches.
Because of the low density of the exosphere and also because the Karman Line, which is 100 km above the Earth, which the line designated where Earth ends and outer space begins, the exosphere is sometimes ignored.
So if one accepts the top of the Thermosphere as being the effective upper boundary of the atmosphere the effective thickness of the atmosphere is only 700 km compared the radius of the Earth, which is 6370 km.