$e^{π\sqrt{163}}$ is almost an integer about $262537412640768744$.
Let $\delta = -\log_{10}{\left|[x] -x\right|}$, where $[x]$ means round $x$.
$\delta(e^{π\sqrt{163}}) \approx 12.125$, I searched for integers less than one million and did not find a larger one.
\begin{array}{l|l} n & \delta \\ \hline 163 & 12.12498077672643 \\ 652 & 9.785848809982161 \\ 1467 & 8.005405448223059 \\ 58 & 6.750100588198867 \\ 2608 & 6.510795060473921 \\ 478233 & 6.336824235216279 \\ 881967 & 6.179002616555315 \\ 880111 & 6.049818696834100 \\ 67 & 5.873691345976712 \\ 160874 & 5.779904075261147 \\ 895056 & 5.744074506024677 \\ 22905 & 5.609682778906657 \\ 674707 & 5.608982629715706 \\ 95041 & 5.548433878234871 \\ 486396 & 5.540950430696813 \\ 343732 & 5.507752498953491 \\ 357711 & 5.478776357709137 \\ 54295 & 5.365357632965891 \\ 613399 & 5.357752793344790 \\ 470884 & 5.281604204081554 \\ \end{array}
$163$ is the biggest in class number 1.
$652 = 4 \times 163$, so worse than $163$.
$1467 = 9 \times 163$, so worse than $163$.
$58 = 232/4$, where $232$ is class number 2.
Then I searched in primes, but seems worse.
\begin{array}{l|l} n & \delta \\ \hline 163 & 12.12498077672643 \\ 67 & 5.873691345976712 \\ 2259143 & 5.594638040601917 \\ 1003957 & 5.559376877231691 \\ 2785649 & 5.506950408472213 \\ 3227963 & 5.506387694695041 \\ 2471017 & 5.031440103309939 \\ 33563 & 4.912614873386113 \\ 719 & 4.894475032853204 \\ 28201 & 4.841231735669607 \\ \end{array}
So, is there a integer that makes $e^{\pi\sqrt{n}}$ closer to an integer than 163?