Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers? Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number?
For example, the real algebraic number $\alpha\in(-1,0)$ satisfying
$$65536\,\alpha^{10}+327680\,\alpha^9+327680\,\alpha^8-655360\,\alpha
  ^7-983040\,\alpha^6+16720896\,\alpha^5\\+20983040\,\alpha^4-655360\,\alpha
  ^3-109155805\,\alpha^2-30844195\,\alpha +16762589=0$$
can be represented as
$$\alpha={_4F_3}\left(\begin{array}c\frac15,\frac25,\frac35,\frac45\\\frac12,\frac34,\frac54\end{array}\middle|\frac1{\sqrt5}\right)-\frac{1+\sqrt5}2.$$
(see Bring radical for details)

Here are answers where I used some particular cases when this representation is possible: [1], [2]. These cases are motivating to try to find a general method applicable to all algebraic numbers.
 A: I'll give you a yes and a no, depending on exactly what you are looking for:

Yes.
See Sturmfels' paper "Solving Algebraic Equations in Terms of $A$-Hypergeometric Series"
Consider a root of $x_0 + x_1 t + \cdots + x_n t^n$ as a function $F(x_0, x_1, \ldots, x_n)$. Then $F$ is an A-hypergeometric function (also known as a GKZ-hypergeometric function), associated to the $A$-matrix
$$\begin{pmatrix}
0 & 1 & 2 & \cdots & n \\ 1 & 1 & 1 & \cdots & 1 
\end{pmatrix}.$$
This result was in some sense known before, but Sturmfels writes these functions down very explicitly. 
See Cattani's lectures for a quick intro to A-hypergeometric functions, including the definitions of the terms used above. This fact is Remark 3.23, and Cattani gives the relevant history. 
I must admit that I never learned how to translate the modern A-hypergeometric language into the classical Gauss vocabulary. Note, however, that A-hypergeometric functions are power series in many variables, not one. And that brings me to:

No. 
Abhyankar has a paper which I wish I understood better; I tried to explain my partial understanding here. Let $F(x_5, x_6)$ be a root of $t^6+x_5 t + x_6=0$. I believe that Abhyankar is showing that $F$ can not be expressed in terms of field operations and holomorphic functions of single variables; it intrinsically has to be a function of two variables. So as long as you stay with $F \left( \begin{smallmatrix} a_1 & a_2 & \cdots & a_k \\ b_1 & b_2 & \cdots & b_{\ell} \end{smallmatrix} \mid z\right)$ and connect this together with field operations, I think Abhyankar is telling you you can never express the roots of a general sextic.

Both of my comments address the question of expressing the roots of $x_n t^n + \cdots + x_1 t + x_0$ as functions of $(x_n, \ldots, x_1, x_0)$ by uniform formulas. I am not addressing the question of whether any particular algebraic number might happen to be equal to a hypergeometric function. I see that you already asked a very hard question along those lines about expressing algebraic numbers in terms of exponentials; I expect that proving anything about hypergeometric functions can only be harder.
