Here is an idea that would appear to meet all of your stated requirements. Now, it doesn't meet other reasonable cryptographical requirements; however you never asked for them.
Here is the idea: we work in an appropriately sized Elliptic Curve group (say, P224) with group size $q$ (which is prime), and pick three generators $F, G, H$ (with unknown relationships; perhaps generated using a Hash2Curve method); and:
$$f(X) = F + X$$
$$g(X) = G + X$$
$$h(X) = H + X$$
These operations obviously commute, and we have $f^i(g^j(h^k(X))) = iF + jG + kH + X$.
Going through your requirements:
If now $f,g,h$, is applied $i,j,k$-times to an input $x$ finding/computing $x$ for given $c = f^i(g^j(h^k(x)))$ should be as hard as possible and with this taking more than $O(|i|+|j|+|k|)$ steps.
I assume that, in this requirement, the attacker doesn't know the values of $i, j, k$ (he does know the relative range). In that case, the best search I can find to verify a value $c$ takes $O( \sqrt{i \cdot j \cdot k } )$ time (assuming $i \cdot j \cdot k < q$, obviously); this is larger than $O(i + j + k)$. This search is done by taking the $0F, 1F, ..., iF$, $0G, 1G, ..., jG$, $0H, 1H, ..., kG$, dividing them into two lists where the sum of any three items in the three lists can be expressed as a sum of two if the items in the list, and then applying a 'big-step/little-step' style algorithm.
Furthermore the methods $f,g,h$ are format-preserving: $X \rightarrow X$, so every output can serve as new input.
As long as you're cool with $X$ being the set of elliptic curve points, we good here.
The max size should be: $|X|<2^{256}$
With P-224, this is true.
Computing $f,g,h$ and their inverses need to take a similar time for each input (independent of $i,j,k$).
We're good here
Furthermore $f,g,h$ have to produce a cycle like $f(f(....f(x)...))=x$ with size $F,G,H$ with $F \approx G \approx H \gg 1$
True; $f, g, h$ all have order $q$, which is much larger than 1
You can easily select ranges for $i, j, k$ so that the target security is met.
Now, the one thing that this idea does not provide is that, given $c, x$ with $c = f^i(g^j(h^k(x)))$, it is trivial to compute $c' = f^i(g^j(h^k(x')))$. However, you never asked that be hard...