In addition to the answer of Harun Siljak one should perhaps mention that for the slightly different expression
$$ \operatorname{dxp}\small([topexponent],[base],[iterationheight])=\operatorname{dxp}(x,t,h) $$
where
$$ \operatorname{dxp}(x,t,0)= x <br>\\\
\operatorname{dxp}(x,t,1)= t^x - 1 <br>\\\
\operatorname{dxp}(x,t,2)= t^{t^x - 1 } - 1 <br>\\\
\cdots
$$
there is a solution for real $h$ based on the power series for the exponential function minus the constant term. Usually this is discussed for the function $$ \operatorname{dxp}(x,e,1)= \exp(x) - 1 $$ and fractional or even irrational heights $h$ and a parametrization for the coefficients for
$$ \operatorname{dxp}(x,t,1)= \sum_{k=1}^\infty u^k{x^k \over k!} <br>\\\
\operatorname{dxp}(x,t,h)= \sum_{k=1}^\infty \mathcal{P}(u,h,k){x^k \over k!} <br>\\\
\text{where I wrote }u \text{ for } \ln(t)
$$ where $\mathcal{P}$ denotes a polynomial in iteration-height, $\ln(t)$ and the series-index $k$.

For series like this and its iterations it is accepted, that the indicated family of iteration heights form a semigroup, where the height-parameter $h$ can be non-integer and can even be complex. This can already be found in L.Comtet's "advanced combinatorics" but also elsewhere.

Unfortunately, although the iterations of *dxp()* and *exp()* can be converted into each other (simply by a change of base) for integer heights, this is not uniquely determined for the fractional heights (the reason is, that for the same base in $b^x$ we have multiple bases $t$ in $t^x-1$ and the various $t$ give different results for the same $x$ and height $h$ if $h$ is fractional). Which then leads to the comment in the other answer, that there is not (yet) a commonly accepted interpretation for the noninteger heights in your original problem.