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This is Buchholz's hydra labeled with ordinals that can be larger than omega.

Modified rule:

1. If $$a$$ has label 0, we proceed as in Kirby-Paris' game. Call the node's parent $$b$$, and its grandparent $$c$$ (if it exists). First we delete $$a$$. If $$c$$ exists (i.e. $$b$$ is not the root), we make $$n$$ copies of $$b$$ and all its children and attach them to $$c$$.
2. If $$a$$ has successor label $$\alpha + 1$$, we go down the tree looking for a node $$b$$ with label $$v \leq \alpha$$ (which is guaranteed to exist, a every child of the root node has label 0). Consider the subtree rooted at $$b$$ — call it $$S$$. Create a copy of $$S$$, call it $$S'$$. Within $$S'$$, we relabel $$b$$ with $$\alpha$$ and relabel $$a$$ with $$0$$. Back in the original tree, replace $$a$$ with $$S'$$.
3. If $$a$$ has transfinite label $$\beta$$, we simply relabel it with $$\beta[n + 1]$$.