What's the value of this Vieta-style product involving the golden ratio?

One way of looking at the Vieta product 2π=2√22+2√√22+2+2√√√2…is as the infinite product of a series of successive 'approximations' to 2, defined by a0=2√, ai+1=2+ai√ (or more accurately, their ratio to their limit 2). This allows one to see that the product converges; if |ai−2|=ϵ, then |ai+1−2|≈ϵ/2 and so the terms of the product go as roughly (1+2−i). Now, the sequence of infinite radicals a0