Läuchli proved in 1961 that $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜2^{2^{\mathfrak{n}}}\cdot2$. As a consequence, $\mathfrak{n}$ is finite if and only if $2^{2^{2^{\mathfrak{n}}}}＜(2^{2^{2^{\mathfrak{n}}}})^2$. It is still open (asked by Läuchli) whether $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜(2^{2^{\mathfrak{n}}})^2$.
Läuchli, H., Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom, Z. Math. Logik Grundlagen Math. 7, 141-145 (1961). ZBL0114.01005.
For an English translation of Läuchli's paper, see here.