The Wikipedia article on Sarkovskii's theorem claims that the Sarkovskii ordering of the natural numbers is not a well-ordering, stating:
Note that this ordering is not a well-ordering, since the set $$\left\{ 2^k \mid k \in \mathbb{N} \right\}$$ doesn't have a least element.
I cannot see how this is true. Surely the element $2^0$ is a least element of this set?
$2^0$ is least in the usual ordering on the natural numbers, but not in the Sarkovskii ordering, which is a different ordering of the naturals. In this ordering, $2^0=1$ is actually the largest natural number, and the smallest is $3$.
$$3<5<\cdots<2^4<2^3<2^2<2^1<2^0$$