Let $a$ be the unique real number such that $a + e^a = 0$. I claim that
(1) $a$ is irrational. (Easy enough: If $a$ were rational, then write $a = p/q$ for integers $p,q$. It follows that $e^a = -a$ is rational, and hence $e^p = (e^a)^q$ is also rational. But this contradicts the fact that $e$ is transcendental.)
(2) $a$ is transcendental. (Is this true?)
(3) $a = -e^{-e^{-e^{-e^{\cdots}}}}$
Anyone know of any other properties of $a$?
The constant you are describing is the negative of the omega constant. That is
$$a=-\Omega=-0.5671432\dots$$
This is easily seen with the identity
$$e^{-\Omega}=\Omega$$
$$\Rightarrow e^a=-a$$
$$\Rightarrow a+e^a=0$$
In answer to your second question, $\Omega$ is known to be transcendental (see here). Another definition for $\Omega$ is the real number such that
$$\Omega e^\Omega=1$$
I'll provide a proof of the third fact just to complete this question: call the power tower $x$
$$x=-e^{-e^{-e^{-e^{\cdots}}}}$$
Then
$$x=-e^{x}$$
$$\Rightarrow x+e^x=0$$
$$\Rightarrow a=x$$
The solution is intersection of $y=x$ and $t=e^x$, which is unique and negative. Let $$f(x)=x+e^x \implies f(0)=1>0, f(-1)=-1+e^{-1}<0$$. so the solution $(x=x_0)$ lies in $(-1,0)$. Numerically it is $x_0==0.5671..$
