Metric independent affine connections

Yes, there is a whole host of other geometric theories of gravity where the connection used isn't the Levi-Civita one. For example, see metric-affine gravity, where the full connection used can be written as something like $$ tag{1} Gamma^{a}{}_{bc} = big{ {}^{,a}_{bc}big} + T^{a}{}_{bc} + frac{1}{2} Q^{a}{}_{bc} , $$
where $big{ {}^{,a}_{bc}big}$ are the Christoffel symbols, $T^{a}{}_{bc}$ is the torsion tensor and $Q^{a}{}_{bc}$ is the non-metricity tensor. By using the full connection you can then find the full Riemann tensor with the usual formula (though you have to be careful with the geometric interpretations of things). From the Riemann tensor you can contract to get the Ricci scalar, which I'll just write schematically of the form

$$tag{2} R = R_{{}} + T + B_T + Q + B_Q ,$$ where $R$ is the Ricci scalar of the full connection whilst $R_{{}}$ is the Ricci scalar associated with the Christoffel symbols. $T$ and $Q$ are contractions of the torsion and non-metricity tensors, and $B_T$ and $B_Q$ are boundary terms associated with them.

In standard GR you pick the Levi-Civita connection with vanishing torsion and non-metricity, so the full connection above is just given by the Christoffel symbols. You then have that $R=R_{{}}$. (A nice reference for this stuff that isn't too technically difficult is Ricci-Calculus by Schouten.)

An approach that has been studied quite a lot is to work with a flat space where the total Riemannian curvature tensor vanishes (with vanishing non-metricity also). Then using (2) you can rewrite $R_{{}}=-T$ and put this into the Einstein-Hilbert action (ignoring boundary terms). This is known as Teleparallel gravity, which is has been studied quite a lot (see the question here Why do we study teleparallel gravity if it is equivalent to general relativity?). You can do a similar thing with a flat geometry and vanishing torsion, but with non-metricity instead - this is sometimes called symmetric teleparallel gravity. I won't go into the detail of either of these but you can go and take a look if it interests you.

The mathematics for some of these theories gets messier for quite a few reasons. Including torsion necessarily brings in issues relating to Lorentz invariance ($T$ itself isn't a Lorentz scalar), and working with non-metricity makes calculations harder. Also, rather than just the metric $g$ being the only dynamical variable, you really need to consider other objects too (this introduces problems relating to how you couple matter to your theory, and new currents analogous to the metric stress-energy tensor).

For the mathematics and a more full overview, the most thorough review is probably the following: Hehl, Friedrich W., et al. "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance." Physics Reports 258.1-2 (1995): 1-171.