Something interesting I noticed while playing around with differential equations…
\[y'=y \rightarrow y=e^x\]\(y'=y^2 \rightarrow y\) has a vertical asymptote
\[y'=y \log y \rightarrow y=e^{e^x}\]\(y'=y (\log y)^2 \rightarrow y\) has a vertical asymptote
\[y'=y \log y \log \log y \rightarrow y=e^{e^{e^x}}\]etc …
similarly
\(\int dx 1/x\) is unbounded
\(\int dx 1/x^2\) is bounded
\(\int dx 1/(x \log x)\) is unbounded
\(\int dx 1/(x (\log x)^2)\) is is bounded
There is no fastest or slowest growing function. But there is some boundary between the infinite iterated logs. As you get closer to the boundary, the solution to the differential equation is an ever faster growing function. Eventually it will surpass \(e^{e^{...}}\) and beyond.