What are examples of non-Levi Civita connections (for example on a sphere)?

Question

Since connections in general do not require that we have a Riemannian manifold, but only a smooth manifold, I find it kind of weird that the only examples of connections that I find in the internet are those which use the Levi-Civita connection.
So, I wanted to know of any examples of non-Levi-Civita connections.
If somebody could also give an example of an easily-visualizable parallel transport (like on $S^2$) using that connection it would be great.