I thought to add an answer instead of giving long comments.

From Wikipedia we have the following quote

"In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that $e$ is not a root of a third degree polynomial with rational coefficients. In particular, $e^{3}$ is irrational."

The reference quoted is Hurwitz, Adolf (1933) [1891]. "Über die Kettenbruchentwicklung der Zahl $e$".

Luckily after much searching I was able to find this reference in an old journal available on internet archive. Here Hurwitz analyzes the simple continued fractions of numbers related with $e$ and notices that most of them have terms in an arithmetic progression (after a certain point).

He then proves the following theorem:

*If simple continued fractions of two positive numbers $x, y$ have terms which are in arithmetic progression (after a certain point) then we can't have a non-trivial bi-linear relation of the form $$y = \frac{Ax + B}{Cx + D}$$ with integer coefficients $A, B, C, D$ unless the terms in their continued fraction belong to the same arithmetic progression.*

Then Hurwitz notes that $$x = \frac{e - 1}{2} = [0, 1, 6, 10, 14, 18,\ldots]$$ and $$y = \frac{e^{2} - 1}{2} = [3, 5, 7, 9, \ldots]$$ where notation $$[a_{0}, a_{1}, a_{2}, \ldots]$$ represents the continued fraction $$a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}}$$ And clearly both of them have terms belonging to arithmetic progressions ($6, 10, 14, \ldots$ and $3, 5, 7, 9, \ldots$ respectively) but these are not the terms belonging to same AP and hence there is no non-trivial bi-linear relation of type $$y = \frac{Ax + B}{Cx + D}$$ with integer coefficients $A, B, C, D$.

Now it follows easily that $1, e, e^{2}, e^{3}$ are linearly independent over $\mathbb{Q}$. If it was not the case then we have integers $a, b, c, d$ not all $0$ such that $$ae^{3} + be^{2} + ce + d = 0$$ Using $e^{2} = 2y + 1$ and $e = 2x + 1$ we get $$a(2x + 1)(2y + 1) + b(2y + 1) + c(2x + 1) + d = 0$$ which leads to $$Axy + Bx + Cy + D = 0$$ with $A, B, C, D$ as integers or $$y = -\frac{Bx + D}{Ax + C}$$ and this is not allowed by the theorem of Hurwitz mentioned above.

Unfortunately I could not understand the proof of his theorem on continued fractions (because the whole paper/journal is in German). With reasonable effort and Google Translate I was able to understand the gist of the paper and I have presented the same in this answer. I have asked for the proof of Huzwitz theorem on MSE.