TL;DR:
JSON does not support Int64 well. The current recommendation is that anyone using the count or capacity values should avoid using extremely large numbers.

Issue with setting count or capacity for Counter to max(int64):

When the Counter capacity or count are set to max(int64) in the game server spec and running kubectl create -f fleet.yaml we get an error:
spec.template.spec.counters.sessions.capacity: Invalid value: "number": spec.template.spec.counters.sessions.capacity in body must be of type integer: "number"

When the Counter capacity or count in the game server spec are close to, but not at, max(int64) kubectl create does not error. However, when checking the details of the game server with kubectl describe the values are truncated:

spec:
  template:
    spec:
      counters:
        sessions:
          count: 8611686018427387904
          capacity: 8611686018427387904

becomes

Spec:
  Template:
    Spec:
      Counters:
        Sessions:
          Capacity:  8611686018427388000
          Count:     8611686018427388000

Debugging:

@zmerlynn root-caused the issue #3608, and found that the source issue is with jsonpatch translating values to scientific notation. A different repo ran into the same error and has a longer description of the issue evanphx/json-patch#189. We're unable to switch to this repo because the fixed method CreateMergePatch just gives back bytes, but in tests, etc. we use the jsonpatch structures. This will likely need to be an upstream fix in gomodules/jsonpatch or kubernetes-sigs/json. The takeaway is that JSON does not support int64 well, and if the controller is having this many issues there are likely other issues that would crop up elsewhere in the code, and other languages would have yet another set of issues.