I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to process in reasonable time with numerical libraries or algorithms known to me — the 1st term of the continued fraction has more than $10^{10^6}$ decimal digits. Is there a trick that allows to do such calculations faster?

UPDATE: I created a sequence A225053 in OEIS for 2nd terms of continued fractions for power towers $e,\,e^e,\,e^{e^e},\,\dots$ Please feel free to extend it if you find a way to calculate further terms.