For gaming, if you must use wifi at all, keep it on a single router. Any additional network hops add latency, any range extender cuts throughput, and any multipath propagation risks packet loss. It's usually better to upgrade a router's antennas to improve the receiver gain (for $5-$20 on eBay). Definitely avoid meshes, since you don't know how many hops the mesh will take and the backhaul links can change as people move around the house.
But if you need to get Internet connectivity from one end of the house to another, and are gaming on a console or PC, why not use a wired connection to minimize latency? You could set up a 1 Gbps Ethernet over coax bridge for less than what a mid-range router would cost. (I just spent the morning...