Is there an NP-hard problem for which no Fixed-Parameter Tractable algorithm exists?
You have to be a bit careful with your question here. Note that an NP-hard problem is a decision problem, while FPT algorithms solve parametrized decision or search problems. So the question is a bit poorly formed. However, I think the question you probably intended to ask is:

Is there an NP-hard problem for which we can add a parameter1 to create a &quot;natural&quot;2 parametrised problem for which no FPT algorithm exists?

To which the answer is (unconditionally!) yes.
First of all, note that FPT, the class of problems that are solvable via a fixed parameter tractable algorithm, is a proper subset of XP, the class of &quot;slice-wise polynomial&quot; parameterized problems that can be solved by a polynomial-time algorithm if the parameter is fixed. In other words: $\mathrm{FPT} \subsetneq \mathrm{XP}$. (I must confess I'm not able to provide the proof by &quot;standard diagonalization&quot; which my source offers as the only justification. Perhaps a complexity theorist can help me out here)
Next, note that since at least one problem in XP cannot be solved by an FPT-algorithm, any XP-hard (in the sense of FPT-reductions) problem cannot be solved by an FPT-algorithm.
In the chapter &quot;Provable Intractability: The Class XP&quot; in Downey and Fellows' Fundamentals of Parameterized Complexity, they complete the argument by showing that what they call the PEBBLE GAME PROBLEM is XP-hard by &quot;reinterpreting&quot; a problem that is known to be at least PSPACE-hard (after &quot;removing the parameter&quot;), so certainly NP-hard. See there book chapter for more details.

Let me add that this result was very surprising to me as well, because for most practical problems, we require al sorts of conjectures ($P\neq NP$, ETH, SETH, 3-SUM, etc.), but this result is an actual fact that is independent of any conjecture.

1: To clarify, by &quot;adding a parameter&quot;, I mean given an NP-hard problem $L\subseteq \Sigma^*$, define a parametrized problem $L'\subseteq \Sigma^* \times \mathbb{N}$ as $L':= \{\langle x, k\rangle \mid f(x)=k\}$ for some function $f : \Sigma^* \rightarrow \mathbb{N}$. This captures the intuitive idea that the additional parameter measures a property of the input.
2: The definition in 1 still allows all sorts of strange parameterizations with functions such as $f(x)\equiv 1$. Ideally, we'd require $f$ to measure something meaningful about the instance, but that seems hard to formalize. I couldn't think of any other formalisation that removes all &quot;unnatural&quot; parameterizations, either. So, I will instead copy the informal notion of &quot;natural parameterized problems&quot; from Downey and Fellows' book.