Well, here's something just to get the ball rolling...

If you use powers of 2, that is , that gives you . Because of the inequality, you can let and for . This gives

Now imagine letting be an arbitrarily large number L, and denote . Now the maximal value of would be gotten by defining , giving it a value of which tends to 1 as L grows large. Thus

I am trying to imagine an induction approach to repeat this last trick on and so on, but as of right now, the highest value I can generate is