The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.
What happens when $n$ is noninteger?
Can we evaluate, e.g. $a[2.5]b$, $a[\pi]$b$, etc?
How about $\frac{\mathrm{d}}{\mathrm{d}x}a[2.5]x$, $\frac{\mathrm{d}}{\mathrm{d}x}x[2.5]b$, or $\frac{\mathrm{d}}{\mathrm{d}x}a[x]b$? Integrals?
Is this a meaningful concept?