Hyper-leviathan number

The hyper-leviathan number is defined like so:

$$\underbrace{|||\ldots|||}_{10^{667}}1\underbrace{|||\ldots|||}_{10^{667}}$$$$_1$$$$\left(10^{666}\right)$$.
 * 1) Let $$|1|_1(x) = x$$.
 * 2) Let $$|1|_n(x) = \prod^{x}_{i = 1} |1|_{n - 1}(i)$$
 * 3) Let $$|2|_1(x) = |1|_x(x)$$.
 * 4) Let $$|2|_n(x) = \prod^{x}_{i = 1} |2|_{n - 1}(i)$$
 * 5) Let $$|3|_1(x) = |2|_x(x)$$.
 * 6) Let $$|3|_n(x) = \prod^{x}_{i = 1} |3|_{n - 1}(i)$$
 * 7) Continue in this fashion. Now define $$||1||_1(x) = |x|_x(x)$$.
 * 8) Let $$||1||_n(x) = \prod^{x}_{i = 1} ||1||_{n - 1}(i)$$
 * 9) Let $$||2||_1(x) = ||1||_x(x)$$.
 * 10) Let $$||2||_n(x) = \prod^{x}_{i = 1} ||2||_{n - 1}(i)$$
 * 11) Let $$||3||_1(x) = ||2||_x(x)$$.
 * 12) Let $$||3||_n(x) = \prod^{x}_{i = 1} ||3||_{n - 1}(i)$$
 * 13) Continue in this fashion. Now define $$|||1|||_1(x) = ||x||_x(x)$$.
 * 14) Let $$|||1|||_n(x) = \prod^{x}_{i = 1} |||1|||_{n - 1}(i)$$
 * 15) Let $$|||2|||_1(x) = |||1|||_x(x)$$.
 * 16) Let $$|||2|||_n(x) = \prod^{x}_{i = 1} |||2|||_{n - 1}(i)$$
 * 17) Let $$|||3|||_1(x) = |||2|||_x(x)$$.
 * 18) Let $$|||3|||_n(x) = \prod^{x}_{i = 1} |||3|||_{n - 1}(i)$$
 * 19) Continuing in this fashion, the hyper-leviathan number is