This is happening because of a security feature called User Account Control (UAC), which Microsoft first implemented in Windows Vista. The UAC feature is meant to allow non-administrative users to run software which requires administrative access to the system, by prompting the user to elevate permissions (e.g., the password prompt you are seeing) only when it is absolutely necessary. These prompts happen even when you are logged on as an administrator because the operating system limits your access unless you explicitly tell it you want to elevate (via a prompt).
There is a good discussion of this issue on a Technet forum (UAC - Stop Asking Me for an Admin Password!), where the solution suggested is either to turn off UAC system-wide or to run the software as an administrative user and turn off UAC for the specific user. Both of these solutions have significant drawbacks, so you would need to decide whether you believe the risk is worth the benefit of avoiding the password prompts.
According to Microsoft (Turn User Account Control on or off):
We don't recommend turning User Account Control off. If you do turn it
off, you should turn it back on as soon as possible.
Unfortunately, up until the release of Windows Vista it was common for software to be developed on the assumption that the user running it would be an administrator, and administrators would have full access to the system at all times. So today there is still a lot of software out there which makes this assumption. The game you are running seems to be developed with this assumption, and the long-term solution to your problem is that the game developer should modify the software to make the user experience better - then Windows won't complain so much and you won't have to make the sorts of changes to your system's security which are not recommended.
There are other ways to address problems like this one, without making significant changes to your system's security, but they usually require a lot of research and configuration at a lower system level than you might be comfortable doing. I don't know the name of the game you are using but searching online for this issue for that particular game title might give you detailed steps.
As an example of what I mean, for a long time Intuit's Quickbooks application would not run without administrator rights. But the solution was very simple - it turns out that it only required access to be able to modify a file inside the Quickbooks folder within
Program Files. By granting an ordinary user access to modify this very limited location, the application would run fine and the system would still be nearly as secure as it was before the change.
In many cases the application you are using really only requires an ability to access a few specific files or registry keys and by opening up the access on these specific system resources you might be able to avoid big security changes. Microsoft provides a tool which can be used to try to determine exactly what access is needed (Process Monitor) but this tool is difficult for most people to use.
Making these sorts of fine-grained changes is not always possible or practical, but it might be worth pursuing either here on Superuser or with the game vendor's support team.