Elementary bounds on parameter in fixed-parameter tractability? In the definition of (strong) fixed-parameter tractability, the time bound is an expression of the form $$f(k).p(|x|),$$ where the input instance is $(x,k)$ with parameter $k$, $p$ is a polynomial, and $f$ is a computable function.

It is possible to replace the computability requirement for $f$ with other classes of functions, as long as the notion of reduction is similarly restricted.  (For instance, Flum and Grohe cover exponential and subexponential families in chapters 15–16 of their textbook, with the associated erf and serf reductions.)


  Has anyone studied the family of elementary functions for the parameter bound $f$?


An elementary function can be bounded above by a fixed tower of exponentials, so this class is closed under composition.  The growth in the parameter in a reduction must then be bounded above by an elementary function as well.

There do exist interesting problems from automata theory which are fixed-parameter tractable, but where the parameter bound is non-elementary (unless P = NP, see Frick and Grohe, doi: 10.1016/j.apal.2004.01.007).  I am wondering if anyone has looked at the fixed-parameter tractable problems which exclude fixed values of the parameter leading to such "galactic" constants (to use Richard Lipton and Ken Regan's term).  Speculating wildly, such a restriction might have useful connections with finite model theory, such as being characterized by a fragment of monadic second-order logic that doesn't lead to the non-elementary constants that can arise from applying Courcelle's Theorem to a fragment with unbounded quantifier alternation.
In his dissertation thesis "Modiﬁzierte parametrische Komplexitatstheorie", Mark Weyer considered, among other things, hierarchies within FPT w.r.t. the function f and reductions between them.   He also did indeed relate these sub-hierarchies to fragments of FO and MSO: Chapter 6 is essentially about the relation between FO/MSO (the number of quantifier alternations of the formulas) and the function f(w) in Courcelle's theorem (w being the treewidth).  He considered both upper and lower bounds and, using the aforementioned reduction framework between certain hierarchies within FPT, he was able to give quite tight bounds.  The examiners of the thesis were Flum and Grohe.

Unfortunately the thesis is in German, and I'm not aware whether material of his thesis has been published in an English journal.  I'm therefore aware that they might be of limited use for you, but nevertheless the references therein might be a good starting point.