Fibonnaci Nested Radicals

For the given nested radical, a real number L exists as the evaluation of the expression.

$2\cdot 2\sqrt { 2\sqrt [ 3 ]{ 2\sqrt [ 5 ]{ 2... } } }$ = L

If the sum of the first $\left\lceil L \right\rceil$ terms of the decimal expansion of L is S, then what is S?

Assumptions:

The nth radical index is the nth fibonacci number. The first two numbers can be written as numbers with a radical of index 1.

Inspired by Brian Charlesworth's problem Harder Nested Radicals