For any $x\in\mathbb{R}^+$, let $x\diamond 1=x$ and $x\diamond (n+1) = x^{x\diamond n}$ for $n\in\mathbb{N}$. For example, $2\diamond 3 = 2^{2^2}=16$.
If $t$ be an unique positive real number such that $t^t = 10^{10^{10}}$, I get $10^{10}>t$.
If $k$ be an unique positive real number such that $k\diamond 50 = 10\diamond 51$,
and $s$ be an unique positive real number such that $s\diamond 51 = 10\diamond 52$.
Is it true that $k>s$ ?
Firstly, $$k>s\iff k\diamond 51>s\diamond 51\iff k^{k\,\diamond\,50}>10\diamond 52\iff k^{10\,\diamond\,51}>10^{10\,\diamond\,51}\iff k>10$$ (assuming $k,s>1$). Now note that $$k\diamond50>10\diamond50\implies k>10.$$