Tetration by a Non-Integer Does anyone think that tetration by a non-integer will ever be defined ... really properly?
Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time to no avail ... and then eventually Leonhard Euler devised a means of doing it, & by a sleight-of-mind that was just so slick & so simple in its essence ... and yet so radical! Is there any scope for another such sleight-of-mind as that, whereby someone might do similarly for tetration, or have they all been used-up? It's impossible to conceive of how there might be any truly new conceptual resource left of that kind. But it's actually tautological that that is so, because the kind  I am talking about is precisely the kind that is essentially radically new, & of even how it might be thought hitherto unconceived!
But the attempts I have seen so far at defining tetration by general real number look to me for all the world like mere interpolation - some of them indeed very thorough & cunning & ingenious (insofar as I can follow them atall) - but lacking that spark of essential innovation that is evinced in Euler's definition of the gamma function.
Just incase it seems I have gotten lost in philosophy, I'll repeat the question: will there ever be a definition of tetration by general real number that resolves the matter as thoroughly as Euler's definition of the gamma-function resolved the matter of factorial of general real number?
 A: We're trying, but it's hard.
A better analogy than the Gamma function would be the way you can now define $x^y$ for real $y$. Why does $x^{3/2}$ make sense? Because I can solve $y^2=x^3$. (Admittedly any solution $y=y_0$ implies $y<y_0$ is a solution too, but we have a convention to get around that for $x>0$.)
So what would ${}^{3/2}x$ mean? Presumably, a solution of $y^y=x^{x^x}$ (although, as @GottfriedHelms notes, we can't expect a solution of ${}^2y={}^3x$ to also satisfy ${}^{2k}y={}^{3k}x$ for all $k\in\Bbb N$). Unfortunately, the values of $y^y$ for $y\in (0,\,\frac{1}{e})$ are repeated again for some $y>\frac{1}{e}$ in a... not particularly simple way, so it's already getting confusing. There are similar headaches when trying to define ${}^{k/(2l)}x$ for $x>0,\,k,\,l\in\mathbb{N},\,2\nmid k$.
(On second thoughts, ${}^42=2^{2^{2^2}}=2^{2^4}=2^{16}$ but ${}^2({}^22)=(2^2)^{2^2}=4^4=2^8$, so there's no ${}^{ab}x={}^a({}^bx)$ rule we can use in the above manner anyway.)
I'm not sure whether you can even prove ${}^yx$ with $y$ irrational can be defined by continuity, i.e. whether we can prove any rational sequence $y_n$ with $\lim_{n\to\infty}y_n=y$ gives the same $n\to\infty$ limit of ${}^{y_n}x$.
Having said all that, I bet we'll have made a lot of progress within 200 years (even if only in proving what we can't do).
A: The idea of E.Schroeder in the 19'th century for bases $b$ (allowing two real fixpoints for iterations, for instance $b=\sqrt{2}$) , sometimes called "regular iteration" seems to me a real good one. It allows a meaningful expression for fractional iteration from some starting point $z_0$ towards some endpoint $z_h$ where $h$ means the (possibly fractional, even complex) iteration-(h)eight, such that $z_0$ is the initial value, $z_1 = b^{z_0}$ is the first (integer) iteration and so on.        
However, that method needs conjugacy to be able to be applied to all real $z_0$ : one has to choose the appropriate fixpoint for shifting the power-series for the exponentiation with base $b$ and get an evaluatable analytical answer at all.
Unfortunately it has been observed, that in the cases $z_0$ where each fixpoint-conjugacy can be taken, the results of fractional iteration are different - even if only by some 1e-25 or so.                  
A basic problem for the general $b$ is the multivaluedness of the complex exponentiation/logarithmization, or say, the "clock-arithmetic" with respect to the $2 \pi î$-term in the exponentiation - I have once seen an article (R.Corless & al.) on the "winding number" which tries to make sense of introduction of one more parameter for the complex numbers to overcome that "clock-arithmetic" to a real-arithmetic. But that was no real progress for the problem here.            
So I think, similarly to the full workout of L. Euler about the multivaluedness of the logarithm and then the representation of the gamma-function as an infinite sum of partial products, we need some more idea here - the E. Schroeder-idea seems to me just like a small insular solution, however nice ever...            
(just as a remark: you might consider the two common versions of the interpolation of the Fibonacci numbers $fib(n)$ to continuous functions of the index: there is one ansatz providing real numbers for real index, and another ansatz (in analogy perhaps to the Schroeder method) which gives complex numbers for real indexes but seems more smooth when seen overall in the complex plane. See a small essay about this at my homepage)
A: There is a remarkable expression I've found in this connection that might have some bearing on the matter for the iterates the Taylor series of $$\operatorname{f}(x)\to x^{\operatorname{f}(x)}$$ when $x\equiv e^z$, such that $$\operatorname{f_0}(x)\equiv1 ,$$$$\operatorname{f_1}(x)\equiv \exp(z) ,$$$$\operatorname{f_2}(x)\equiv \exp(z\exp(z)) ,$$ etc. The coefficients for $k=0\dots n$ are those of the Lambert W-function; but thereafter, for $k>n$ the coefficients are given by the following recursion. Let $a_{n,0}=1\forall n$, $a_{0,k}=0$ for $k>0$, & thereafter $$a_{n,k}={1\over n}\sum_{j=1}^k ja_{n,k-j}a_{n-1,j-1} .$$ These coefficients are in a sense 'wasted' upthrough $k=n$, inthat they do not actually appear in the Taylor series; and yet they still serve the function of being necessary for the generation of the coefficients that do appear. It is fascinating to my mind, the way there is a kind of discontinuity in the series - the coeffiecients generated by this recursion 'peeling-away' one-at-a -time as $n$ is incremented, 'revealing' the coefficients of the Lambert W-function 'underneath'; and it is a well known result that the limit as the $$\operatorname{f}(x)\to x^{\operatorname{f}(x)}$$ tends to $\infty$ is indeed the Lambert W-function.
Whether this is susceptible of treatment by Schroeder's method I would not venture definitely to say at the present time, as, though I see in outline how that method can be applied to something like the recursion that gives the Fibonacci numbers, I am rather daunted by that discontinuity in the generation of the coefficients in this case; and I cannot see at a glance how it would be encoded.
