How much hydrogen is in one gallon of water?

Answer

If you mean hydrogen gas, H₂, just naturally present in the water, then essentially none.

If you mean how much hydrogen gas could be generated by the electrolysis of 1 gallon of water, that is a stoichiometry problem. The balanced reaction is:
2H₂O --> 2H₂ + O₂
The mass of 1 gallon of water is 3.7854 kilograms, which is 210.3 moles of H₂O. From the stoichiometry, there is a 1 to 1 (2 to 2) ratio of water to hydrogen produced, and so 210.3 moles of H₂ will be produced. 210.3 moles of H₂ weighs 420.6 grams, or just under one pound (see the Related Questions links to find out how to calculate the volume of that much H₂ produced).

If you mean hydrogen ions, H⁺, then it depends on the pH of the water. In perfectly pure, neutral water with pH of 7, then the concentration of H+ is 10^-7 moles per liter. In one gallon, there are 3.7854 liters. So in one gallon of pure water, there are 3.7854 * 10^-7 moles of H⁺.

See the Related Questions for more information about pH and how it measures the concentration of H⁺.