Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?

One thing you can try is to select numerical values for $a$ and $b$ (say $a=b=1$ or $a=b^{-1}=2$) and then find a power series solution $y=1+\sum_{k>0}c_k(x-1)^k$ and inspect the coefficients $c_k$. With $a=b=1$ they are $$ 3, 2, 4, 11, 35, \frac{721}{6}, \frac{18163}{42}, \frac{540391}{336}, \frac{98091}{16}, \frac{26684211}{1120},\dotsc $$ If there was a straightforward solution in terms of standard functions then you would expect that the coefficients would be arithmetically simple, with no large primes involved in $c_k$ when $k$ is small. (Large primes can appear in the numerator by accident, but not so easily in the denominator.) Also, the terms $k!c_k$ seem to be integers, so we can look them up in the Online Encyclopedia of Integer Sequences. In fact, the prime $286927$ appears in the numerator of $c_{10}$ and OEIS does not recognise the sequence $k!c_k$ so that is not promising for the existence of an elementary solution.

Maple does not find a closed-form solution. Note that the substitution $y = 1/u$ produces the Abel DE $$u' = - a u^3 - \frac{b+x}{x} u^2$$ but Maple can't find a closed-form solution for that either: it doesn't seem to fall into one of the known exactly solvable classes.

EDIT: You might also note that scaling the dependent and independent variables by $y = cY$, $x = cX$ gives you a similar equation for $Y(X)$ with $a$ and $b$ replaced by $a/c$ and $b/c$ respectively. So effectively there is just one parameter here.