After seeing the problem of which is bigger out of $9^{10}$ or $10^9$ (and eventually working out a few ways to answer that) I got interested in where it switches. I.e.
$2^3$ < $3^2$
But:
$3^4$ > $4^3$
So, $x^{(x+1)}$ = $({x+1})^x$ must be a value for $x$ between $2$ and $3$. Naturally I guessed $e$, but that is too high.
I've tried taking logs of each side, to either base $x$ or base $x+1$ but I keep ending up stuck. E.g. $x+1 = x \log_x(x+1)$ and the only thing I know to do with $\log(a + b)$ is turn it into a $\log(a) + \log(1 + b/a)$, which doesn't get me anywhere.
So, being a pragmatic guy, I plotted $x^{x+1}-({x+1})^x$, and started thrashing around trying to find an equation to match where it crosses the $x$-axis. This is really close - slightly too large (2.29316638129); change the $7$ to $8$ and it is slightly too small ($2.29316607539$):
$$x=e-\left(\frac{1}{e}\right)-\left(\frac{1}{e^3}\right)-\left(\frac{1}{e^5}\right)-\left(\frac{2}{e^8}\right)-\left(\frac{2}{e^{11}}\right)-\left(\frac{2}{e^{13}}\right)-\left(\frac{7}{e^{15}}\right)$$
But this stinks of over-fitting, doesn't it. :-)
You are playing with an old toy :)
The constant you are looking for is called Foias-Ewing constant and the exact value is:
You can find more about this number and its properties here:
Honestly I did not know anything about this number 5 minutes ago. I just Googled a few starting digits of your number and... voila! :)