What happens when I run rpm-ostree upgrade multiple times without rebooting?

Upgrades count only if you reboot, so booting your install with the latest layer/snapshot (unless you use rpm-ostree ex livefs).

From the Red Hat docs (https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux_atomic_host/7/html/installation_and_configuration_guide/managing_atomic_hosts):

If you run rpm-ostree install several times consecutively without rebooting or applying changes live, only the most recent command will take effect. If you install wget and strace consecutively and reboot afterwards, only strace will be present after reboot.

I assume the same is valid for the upgrade. You got me curious about this…