User blog comment:Alemagno12/Graham Number Systems/@comment-25337554-20151214092441/@comment-27173506-20151214093340

I think you mean that the first sequence is approximately

3->3->1->3, 3->3->2->3, 3->3->3->3...

Since we're iterating nestings of up arrows, not up arrows themselves.

The second sequence would be approximately

3->3->1->4, 3->3->2->4, 3->3->3->4

as we're iterating the previous sequence.

In general, the "k"th member of sequence number n would be approximately:

3->3->k->n+2

This means it caps out at f_w2.

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As @KthulhuHimself stated, it is similar to Aarex's Graham Generator, and the early levels of my GoogolX notation are based upon the same lines, but I hope it develops into something interesting.