Maclaurin Expansion for $e^{e^{z}}$ at $z=0$ I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$.  
I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms into the Taylor series expansion for $e^{z}$ as follows:

$e^{\omega} = \sum_{n=0}^{\infty}\frac{\omega^{n}}{n!} = 1 + \omega + \frac{\omega^{2}}{2!}+\frac{\omega^{3}}{3!}+\cdots = 1 + (1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\frac{z^{4}}{4!}+\cdots) + \frac{1}{2}\left(1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\frac{z^{4}}{4!}+\cdots \right)^{2}+ \frac{1}{3!}\left( 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\frac{z^{4}}{4!}+\cdots\right)^{3}+\cdots$

But, then when I try to multiply this all out (using Maple), I find that going even further out than I have here, the terms of degree $\leq 5$ do not terminate. So, then I tried putting the command into Maple, and it came out to be something long and horrible with $e$ coefficients for each term.
I must be missing something here.
Could somebody please tell me how to finish this problem? I would ask for a complete solution, not hints or leading questions. I've been working on this for hours and am beyond frustrated. The most edifying thing for me at this point would be to see the correct, full way it's supposed to be done, work backwards, take it apart, and then be able to apply it to other situations.
Thank you.

Your approach works, but the Taylor series for the outer function should not necessarily be centered at $0$.  Rather, it should be centered at the constant term of the Taylor series of the inner function.
In your example, the outer series should be centered at $x=1$.  Expanding $e^x$ in a Taylor series around this point gives
$$ e + e(x-1) + \frac{e}{2}(x-1)^2 + \frac{e}{6}(x-1)^3 + \frac{e}{24}(x-1)^4 + \frac{e}{120}(x-1)^5 + \cdots $$
Now substitute in the Taylor series for $e^z$ centered at $z=0$ and you get
$$ e + e \left(z+\frac{1}{2}z^2 + \frac{1}{6}z^3 + \frac{1}{24}z^4 + \frac{1}{120}z^5 + \cdots \right) + \frac{e}{2} \left(z+\frac{1}{2}z^2 + \frac{1}{6}z^3 + \frac{1}{24}z^4 + \frac{1}{120}z^5 + \cdots \right)^2 + \frac{e}{6} \left(z+\frac{1}{2}z^2 + \frac{1}{6}z^3 + \frac{1}{24}z^4 + \frac{1}{120}z^5 + \cdots \right)^3 \\ + \frac{e}{24} \left(z+\frac{1}{2}z^2 + \frac{1}{6}z^3 + \frac{1}{24}z^4 + \frac{1}{120}z^5 + \cdots \right)^4 + \frac{e}{120} \left(z+\frac{1}{2}z^2 + \frac{1}{6}z^3 + \frac{1}{24}z^4 + \frac{1}{120}z^5 + \cdots \right)^5 + \cdots $$
As you expand each exponentiated series, you will only find finitely many terms of each given degree.  Doing this and combining like terms, you get your series
$$ e + ez + \left(\frac{e}{2} + \frac{e}{2} \right) z^2 + \left(\frac{e}{6} + e + \frac{e}{6} \right) z^3 + \left(\frac{e}{24} + \frac{e}{6} + \frac{e}{8} + \frac{e}{4} + \frac{e}{24} \right) z^4 + \left( \frac{e}{120} + \frac{e}{12} + \frac{e}{24} + \frac{e}{12} + \frac{e}{8} + \frac{e}{12} + \frac{e}{120} \right) z^5 \cdots $$
which simplifies to
$$ e + ez + ez^2 + \frac{5e}{6} z^3 + \frac{5e}{8} z^4 + \frac{13e}{30} z^5 + \cdots$$
