How high can it be

A real solution exists when the curves $f(x) = a^x$ and $g(x) = x$ share a common tangent with each other. Setting the derivatives of both equal to each other produces:

$a^{x}ln(a) = 1 \Rightarrow x = log_{a}(\frac{1}{ln(a)})$

We now require the points $(x, f(x))$ and $(x, g(x))$ to coincide, or $f(x) = g(x) \Rightarrow \frac{1}{ln(a)} = log_{a}(\frac{1}{ln(a)}) \Rightarrow a^{\frac{1}{ln(a)}} = \frac{1}{ln(a)}$

which is satisfied at $a \approx 1.44467$ (per Wolfram Alpha). Thus the maximum value of $\lfloor{1000a} \rfloor$ equals $\boxed{1444}.$

The maximum value of $x^{\frac{1}{x}}$ is $e^{\frac{1}{e}}$ , where $e=2.71828...$ is the Euler's number. Therefore the required answer is $\boxed {1444}$ .