$f(f(n))=g(n)$

I understand the problem a bit more now, it should say something like: for determined or fixed $g(x)$ with $g(1) = k$ . i.e if $g(x)$ is fixed from this point onwards you can determine infinitely many values of $f(x)$ from $g(x)$ . But this has two solutions: $3, 4$ .

If we restrict $g(x)$ to be linear in $x$ , and state all values of $f(x)$ are uniquely determinable, this has the unique answer of $k = 3$ .

---

$g(1) = k \implies f(1) = a$ and $f(a) = k$ . If $k \geq 3, a = 2$ or $3$ are both valid.

$k = 1, 2$ lead to contradictions of strictly increasing $f(x)$ .

$k = 4$ fixes $f(1) = 2, f(2) = 4$ . This leads to $f(4) = g(2)$ . This is a valid answer: eg $g(x) = 4x$ , determines $f(2^n) = 2^{n+1}$ . (It uniquely determines $f^m(1) \forall m$ . (Note that $f(1) = 3 \implies f(3) = 4 \implies f(2)$ doesn't exist )

$k = 3$ fixes $f(1) = 2, f(2) = 3$ .

---

Assuming $k = 3$ , and $g(x) = ax + b$ (linear):

$g(x) = ax + (3 - a)$ .

$f(1) = 2$

$f(2) = 3$

$f(3) = g(2) = a + 3$

We can determine $f(a + 3) = g(3) = 2a + 3$ .

Also, $f(2a + 3) = a^2 + 2a + 3, f(a^2 + 2a + 3) = 2a^2 + 2a + 3$ .

..

Let $S_n = 2(a^n + a^{n-1} + ... + a + 1) + 1, T_n = S_n + a^{n+1}$ .

Further, $f(S_n) = g(T_{n-1}) = T_n$ and $f(T_n) = g(S_n) = S_{n+1}$ $\forall n$ .

Noting that $T_n - S_n = a^{n+1} = S_{n+1} - T_n$ , and $f(x)$ is strictly increasing, we can populate $f(S_n + c) = T_n + c$ for $c < a^{n+1}$ .

Now we need to populate the values of $f(x)$ for $x$ between $T_n$ and $S_{n+1}$ :

$f(T_n + c) = S_{n+1} + c*a$ , for $c < a^{n+1}$ .

$T_n + a^{n+1} = S_{n+1}$ . Hence we have populated \f(x)) for all $x$ between $S_n$ and $S_{n+1} \forall n$ . Which is all values of $x$ .

I don't know what this question is supposed to mean. I don't know what "concretely determinable" is supposed to mean. There appear to be far too few constraints here to determine infinitely many values of f. It is pretty easy to show that $k \geq 3$ . It's also easy to show that, with no other information, if $k \geq 4$ , then $f(1)$ is not uniquely determined. I cannot see what else is supposed to be going on here.