How big is 10^100^100?

Elliot Temple posed the following math puzzle: How many square roots does it take to get 10¹⁰⁰¹⁰⁰ down to a small number? (And what number is it?)
Note that a^bc = a^(bc).
Answer: 665 square roots takes you down from 10¹⁰⁰¹⁰⁰ to around 4.5.
Proof:
Here, lg means the base 10 logarithm, and dot (·) means multiplication. I will use the following facts:

• (a^b)^c = a^bc
• x^ax^b = x^a+b
• x = 10^{lg x}

First, note that 10¹⁰⁰¹⁰⁰= 10¹⁰²⁰⁰ . We next prove the following lemma:

a^bc = 10^{10c lg b + lg lg a}

Example: 2²³ = 2⁸ = 256 = 10^{103 lg 2 + lg lg 2}
Proof:

	abc
=	(10lg a)(10lg b)c
=	(10lg a)10c lg b
=	10lg a · 10c lg b
=	1010lg lg a · 10c lg b
=	1010c lg b + lg lg a ▢

We can now use the lemma to show that 4.5²⁶⁶⁵ = 10^{10665 lg 2 + lg lg 4.5}. And 665 lg 2 + lg lg 4.5 ≈ 200.▢
You can also check the answer with Robert Munafo's Hypercalc, an online calculator that works with very large integers. To do this, enter 4.5^(2^665) in Hypercalc. It evaluates it to something very close to 10^10^200.