There is not such a thing as a thermodynamic limit to hydrogen removal. Theoretically, all hydrogen can be removed down to the last atom. The point is, however, that the derivative of the Gibbs free energy of a solution with respect to the concentration of a solute tends to infinity as the concentration of the solute tends to zero. Since this derivative represents the driving force to the absorption of the solute, it becomes increasingly more difficult to remove a solute from a solution as the amount of the solute decreases.

This is why it is so difficult to purify any solution from a solute that has any tendency to dissolve in it. One may  try and modify the solution composition to make it repellent to the unwanted solute. The problem is always a question of chemical equilibrium, and it depends on pressure and temperature. But, again, one cannot remove all traces of a component, although different solution compositions or different values of pressure and temperature may mean different equilibrium concentrations of it.