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In his article Transcendental continued fractions, Journal of Number Theory 13, November 1981, 456-462, Gideon Nettler shows that two numbers given by continued fractions $A = [a_0,a_1,a_2,...]$ and $B = [b_0, b_1, b_2, ...]$ have the property that $A$, $B$, $A \pm B$, $A/B$ and $AB$ are irrational if $\frac12 a_n > b_n > a_{n-1}^{5n}$ for sufficiently large $n$, and transcendental if $a_n > b_n > a_{n−1}^{(n−1)^2}$ for sufficiently large $n$. The growth of the $a_i$ in the continued fraction expansion of $e$ is so small that present methods seem useless for proving the transcendence of $e$ in this way.

Edit. Similarly, Alan Baker proved in Continued fractions of transcendental numbers (Mathematika 9 (1962), 1-8) that if $q_n$ denotes the denominator of the $n$-th convergent of a continued fraction $A$, and if $$ \lim \sup \frac{(\log \log q_n)(\log n)^{1/2}}{n} = \infty, $$ then $A$ is transcendental.

Edit 2 I guess that the answer to your question should be a firm "yes" after all. In

  • Über einige Anwendungen diophantischer Approximationen, Abh. Preu\ss. Akad. Wiss. 1929; Gesammelte Abhandlungen, vol I, p. 209-241

Siegel proved that all continued fractions $$ \frac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \ldots}}} $$ in which the $a_i$ form a nonconstant arithmetic sequence are transcendental. Applying this to the continued fraction expansion of $\frac{e-1}{e+1}$ gives the transcendence of $e$.

Siegel's proof uses, predictably, analytic machinery (solutions of Bessel and Riccati differential equations) going far beyond Liouville's theorem.
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In his article Transcendental continued fractions, Journal of Number Theory 13, November 1981, 456-462, Gideon Nettler shows that two numbers given by continued fractions $A = [a_0,a_1,a_2,...]$ and $B = [b_0, b_1, b_2, ...]$ have the property that $A$, $B$, $A \pm B$, $A/B$ and $AB$ are irrational if $\frac12 a_n > b_n > a_{n-1}^{5n}$ for sufficiently large $n$, and transcendental if $a_n > b_n > a_{n−1}^{(n−1)^2}$ for sufficiently large $n$. The growth of the $a_i$ in the continued fraction expansion of $e$ is so small that present methods seem useless for proving the transcendence of $e$ in this way.

Edit. Similarly, Alan Baker proved in Continued fractions of transcendental numbers (Mathematika 9 (1962), 1-8) that if $q_n$ denotes the denominator of the $n$-th convergent of a continued fraction $A$, and if $$ \lim \sup \frac{(\log \log q_n)(\log n)^{1/2}}{n} = \infty, $$ then $A$ is transcendental.

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In his article Transcendental continued fractions, Journal of Number Theory 13, November 1981, 456-462, Gideon Nettler shows that two numbers given by continued fractions $A = [a_0,a_1,a_2,...]$ and $B = [b_0, b_1, b_2, ...]$ have the property that $A$, $B$, $A \pm B$, $A/B$ and $AB$ are irrational if $\frac12 a_n > b_n > a_{n-1}^{5n}$ for sufficiently large $n$, and transcendental if $a_n > b_n > a_{n−1}^{(n−1)^2}$ for sufficiently large $n$. The growth of the $a_i$ in the continued fraction expansion of $e$ is so small that present methods seem useless for proving the transcendence of $e$ in this way.