Is there an NP-hard problem for which we can add a parameter1 to create a "natural"2 parametrised problem for which no FPT algorithm exists?
- The adding a parameter is needed because a NP-hard problem is normally just a question with a yes or no answer, if you want to limit some parameter you need to specify which one (even though something like $k$-Coloring might have an obvious one already), so with "specifying which parameter" one is limiting, one is "adding a parameter" to the problem. A more detailed description is included in the answer by Discrete Lizard.
- I think Natural tries to exclude "trivial" parameterizations as I discuss in my first doubt in this question. Again a more detailed description is included in the answer by Discrete Lizard.
- It might be a trivial question as it perhaps is possible to always "stuff" the entire problem within the $f(k_1,k_2,..,k_m)$ part of the $f(k_1,k_2,..,k_m)n^c$ algorithm whilst setting $n=c'$ where $c'$ is an arbitrary constant. But perhaps the exact definition of FPT prevents such (ab)use of the concept of FPT.
Based on the comment of plop there indeed exists a trivial way to parameterize "any" (I assume any properly well-posed problem) problem, such that its parameterization is fpt. Those parameterizations use languages, which I assume to be what is described here. Such a "trivial" (in light of the question, not in light of difficulty) parameterization is intended to be ignored. So in the "words" of Discrete lizard: non-trivial parameter range(s) is(are) intended.