User blog comment:! cook the lettuce/Despacit's Number Competition/@comment-37485018-20191208215742/@comment-37485018-20191208223828

Explanation:

Each chain can also be viewed as a rooted binary tree (for example, 0 is a single node, \([a,b]=\overset{a\backslash~/b}{\circ}\)). Because it's a binary tree, \(2Z-1=N\), where Z is the number of 0s in the chain and N is the number of nodes on its tree.

Follow the comparison algorithm for chains written above to determine if \(a<b\) for chains \(a\) and \(b\).

My number is the length of the longest possible sequence of chains where:


 * 1) The sequence starts with \(\underbrace{[[\cdots[[}_{10^6}0,0],0]\cdots,0],0]\)
 * 2) Each chain is less than the last ("less than" using the algorithm to determine if the statement "a<b" is true)
 * 3) The ith chain in the sequence has at most \((10^6+i)!\) 0s in it (alternatively, its tree has at most \(\frac{(10^6+i)!+1}{2}\) nodes)