Function or bounds for the number of solutions of $\sum_{i=0}^k \frac{1}{x_i} = 1$ Is there any result known for the number of different solutions of $1 = \sum\limits_{k=0}^n \frac{1}{x_k}$ in dependency of the length $n$ of this partition?
All I know, up to now, is that there are for every $n$ only finitely many different solutions and the maximal $x_k$ is given by the $k$th element of the Sylvester-sequence.
Further, there is a result by Hofmeister/Stoll about the number of solutions up to length $n$ for $\frac{a}{b} = \sum\limits_{k=0}^m \frac{1}{x_k}$ with $m \leq n$ and $a \neq b$. So unfortunately this is not usable in my case too.
 A: An upper bound of about $c_0^{2^k}$ follows by elementary induction, (from your comment $x_k$ is bounded by the Sylvester sequence).
Here $c_0 \approx 1.264$ is $\lim u_n^{\frac{1}{2^n}}$, ($u_n$ the $n$-th term  of the
Sylvester sequence).
An upper bound of $c_0^{(1+\epsilon) 2^{k-1}}$ is in a paper by C.
Sándor. Sándor also gives a lower bound: for $k \geq 3$:
$\exp(c \frac{k^3}{\log k })$, for some positive constant $c$. His paper is:
Periodica Mathematica Hungarica
Volume 47, Numbers 1–2, 215–219.
On the number of solutions of the Diophantine equation $\sum_{i=1}^n\frac{1}{x_i}=1$
An upper bound of $c_0^{(\frac{5}{24} +\epsilon) 2^{k-1}}$
was proved by T. Browning and myself in our paper
"The number of representations of rationals as a sum of unit fractions", which is to appear in the Illinois J of
Mathematics (vol. 55, no. 2, 685-696, 2011). It is online here:
https://www.math.tugraz.at/~elsholtz/WWW/papers/papers33FINAL2013.pdf.
This paper gives an essentially best possible answer for a general fraction $\frac{m}{n}$ and $k=2$, then a nontrivial upper bound for $k=3$, and as a corollary lifts these results to general $k$. The case $\frac{m}{n}=1$ then follows.
All in all, there is a large gap between upper and lower bound for your original question.
A: The first $8$ terms (under the assumption $x_1\le\cdots\le x_n$) are given at http://oeis.org/A002966 (without the restriction, it's http://oeis.org/A002967). 
There is no information there about exact formulas or asymptotics, which suggests to me that nothing is known (but I haven't checked the Elsholtz paper that Greg Martin mentions). 
There is some discussion of the problem at D11 in Guy, Unsolved Problems In Number Theory, especially page 257 (of the 3rd edition), but the main piece of information there is the attribution to Erdos of the question of asymptotic behavior. 
