I know that value of hypergeometric function at $z = 1$ is given by $$ F(a,b;c;1) = \frac{\Gamma(c)\Gamma(c - a - b)}{\Gamma(c - a)\Gamma(c - b)}, $$ (according to the Digital Library of Mathematical Functions) when $\Re(c) > \Re(a + b)$.
My question: is there an expansion (again assuming that $\Re(c) > \Re(a + b)$) for $F(a,b;c;z)$ in possibly non-integer powers of $z-1$ which has $F(a,b;c;1)$ as the first term?
Also if an expansion like this exists what are the first few terms?
The hypergeometric differential equation with $F(a,b,c,z)$ as a solution near $z=0$ also has solutions at the other two singluar points, $z=1$ and $\infty$. The function $F(a,b,c,z)$ near $z=1$ is a solution of that DE, so we can write all solutions of it as linear combinations of the two hypergeometric solutions $$ {\mbox{$_2$F$_1$}(a,b;\,a+b+1-c;\,1-z)}, \\ \left( 1-z \right) ^{c-a-b}{\mbox{$_2$F$_1$}(c-b,c-a;\,1+c-a-b;\,1-z)}. \tag1$$
The coefficients to use (the "connection coefficients") for your question were worked out by Gauss (or perhaps Goursat?)
Here it is. \begin{align} {\mbox{$_2$F$_1$}(a,b;\,c;\,z)}&={\frac {\Gamma \left( c \right) \Gamma \left( c-a-b \right) {\mbox{$_2$F$_1$}(a,b;\,a+b+1-c;\,1-z)}}{\Gamma \left( c-a \right) \Gamma \left( c-b \right) }} \\\qquad &+{\frac {\Gamma \left( c \right) \Gamma \left( a+b-c \right) \left( 1-z \right) ^{c- a-b}{\mbox{$_2$F$_1$}(c-b,c-a;\,1+c-a-b;\,1-z)}}{\Gamma \left( a \right) \Gamma \left( b \right) }} \end{align}
This is for almost all $a,b,c$ not nonpositive integers. Exceptional cases may occur when $a+b-c \in \mathbb Z$; then the second solution may behave logarithmically near $z=1$. (See this question .)
Reference. Corollary 2.3.3 in
Andrews, George E.; Askey, Richard; Roy, Ranjan, Special functions, Encyclopedia of Mathematics and Its Applications. 71. Cambridge: Cambridge University Press. xvi, 664 p. (1999). ZBL0920.33001.