Rank-Wise Approximations of Hyper-Operations

[Catullus]

Could similar uniqueness criterion work for pentation and higher hyper-operations?

They work for the iteration of any function, so yes.

If you did that to do tetration, pentation, et cetera. How can you, when the hyperoperations converge as the rank becomes larger and larger, approximate the convergence?
2[1]2+1/1 = 5. 2[2](2+1/2) = 5. 2[3](2+1/3) ~ 5.040. 2[4](2+1/4) ~ 5.051. Do they converge? If so what do they converge to? How can you approximate the convergence in a way, such that it becomes better and better as the rank of the hyperoperation increases?