  I present an asymptotic formula for the Takeuchi numbers $T_n$. In
particular, I give compelling numerical evidence and present a heuristic
argument showing that $$T_n\sim C_T B_n\exp{1\over2}{W(n)}^2$$as $n$ tends to
infinity, where $B_n$ are the Bell numbers, W(n) is Lambert's $W$ function, and
$C_T=2.239...$ is a constant. Moreover, I show that the method presented here
can be generalized to derive conjectures for related problems.
