Using the binomial theorem, we get $$ \begin{align} \left(x+\frac1{x^x}\right)^x-x^x &=x^x\left[\left(1+\frac1{x^{x+1}}\right)^x-1\right]\\\ &=x^x\left[\frac1{x^x}+\frac{x-1}2\frac1{x^{2x+1}}+O\left(\frac1{x^{3x}}\right)\right]\\\ &=1+\frac1{2x^x}+O\left(\frac1{x^{x+1}}\right) \end{align} $$ If you are getting wild oscillations or quantized output, it is probably due to truncation error.
The bottleneck actually seems to be in the computation of $x+\frac1{x^x}$ since IEEE double precision arithmetic only has a $53$ bit mantissa. $13$ has $4$ bits and $13^{-13}$ has $48$ zeros after the binary point before the first non-zero bit. So there is just barely enough precision to note that there is a difference between $x+\frac1{x^x}$ and $x$. Any imprecision in the computation would completely overwhelm this difference and cause extreme problems in the final computation.