TITLE: Do operations generate well-ordered sets only?
QUESTION [10 upvotes]: I've read   @TauMu's question   about the set of functions   $\mathbb N\rightarrow\mathbb N$   generated from the identity map by repeatedly applying exponentiation of two already accepted functions. @TauMu asks if such a set is well-ordered with respect to eventual domination.
It seems to me that it would be hard, perhaps impossible, to obtain a set which is not well-ordered regardless of the choice of a binary operation   $\mathbb N\times\mathbb N\rightarrow\mathbb N$.   Here is a patient formulation of my more general question:
Given an operation   $\tau : \mathbb N^2\rightarrow\mathbb N$,   let   $\bigcirc^{\tau}\subseteq\mathbb N^{\mathbb N}$   be the smallest set such that it has the identity map   $I_{\mathbb N}$   as its element,   $I_\mathbb N\in\bigcirc^\tau$,   and   $h:=\tau\circ(f\triangle g)\in\bigcirc^{\tau}$   for every   $f\ g\in\bigcirc^{\tau}$.
REMARK (an explanation of the notation above)
$$\forall_{n\in\mathbb N}\quad h(n) := \tau(f(n)\ g(n))$$
Finally,
QUESTION (edited twice after the 1st and 2nd comment by @Joseph Van Name)   Does there exists an operation   $\tau:\mathbb N^2\rightarrow \mathbb N$   dominating the identity in each variable (see below), and such that   $\bigcirc^{\tau}$   contains an infinite strictly decreasing sequence with respect to the relation of eventual domination?
By   $\tau$   dominating the identity in each variable I mean that:
$$\forall_{k\ n\in\mathbb N}\quad \tau(k\ \ n)\ \ \ge\ \ \max(k\ \ \ n)$$

REPLY [4 votes]: I think the following example shows that the extra condition (nondecreasing in all arguments) proposed in Emil Jeřábek´s answer is necessary.
Consider the operation $$\tau(m,n)=2 \max(m,n)^n -\min(m,n).$$
It is clear that $\tau(m,n) \geq \max(m,n)$. Now let $g_0=I_\mathbb{N}$, $g_{i+1}=\tau\circ(g_i \triangle g_0)$, $h=\tau\circ(g_1 \triangle g_1)$ and $f_i=\tau\circ(g_i \triangle h)$. Then $\{f_i: i \in \mathbb{N} \}$ is a strictly decreasing sequence with respect to eventual domination.