Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$)
Examples [] and [[],[],[]] and [[[]],[[],[],[]],[[],[],[[]]]] but it's much better to think of them like this (can ignore the colors for now) 
I was wondering what is the rank of the lexicographic order on trees?
Formally, we will write a:b for a list of trees whose first element is r and rest is b (r is red in the diagram and b is blue)
$(h:t) <_T (h':t')$ when
- $h=h'$ and $t <_T t'$
- $h <_T h'$ and $t$,$'t$ can be anything (e.g. $t$ could be enormously large compared to $t'$)
and this relation is well founded (I claim)
It is clear that the rank is bigger than $\omega$ since you can embed the naturals into 0=[], 1=[[]], 2=[[[]]], 3=[[[[]]], ... just long chains.
You easily embed $\omega+1$ because $[[],n] <_T [[[]],[]]$ for $n$ corresponding to any natural number tree - but by a similar idea we easily embed $\omega^2$. It seems like the rank of this is large.
Note, this is the same as the relation on http://math.andrej.com/2008/02/02/the-hydra-game/ and I expect has smaller rank.
I am not very comfortable with these notions, so I want to get a rigorous proof in the end - but for now I don't even know the answer.
Are you sure this relation is well founded? For clarity, let $\bot = []$, the empty list.
Consider the following sequence:
\begin{align} T_0 &= \big[[\bot,\bot],\bot\big] \\ T_1 &= \big[\bot,T_0\big] = \Bigg[\bot,\bigg[[\bot,\bot],\bot\bigg]\Bigg]\\ &\ \vdots \\ T_k &= \big[\bot,T_{k-1}\big] \end{align}
Assuming that $\bot <_T [\bot,\bot]$ (you haven't defined this case), we have $$T_1 <_T T_0,$$ that is $$\bot : T_0 <_T [\bot,\bot] : \bot,$$ hence $$T_k <_T T_{k-1}$$ and we have an infinite strictly decreasing sequence.
I hope this helps ;-)