Sounds like you may have a faulty switch, but here are some steps I would try:
1. (In case you haven't already), check to see if there's something close to your monitor that could be pressed against the menu button.
2. Reset your monitor to factory settings. This can usually be done by opening your monitor's OSD menu and selecting "Reset all Settings." Or, unplug your monitor from your computer and hold the power button for 30 - 60 seconds. This should also reset your screen.
3. Lock the screen. Depending on the source of this problem, locking the screen may keep the OSD menu from appearing. Locking the screen is usually done by holding the menu button for 10 - 15 seconds. A padlock or similar icon may appear once it is locked.
If the problem is the result of a faulty switch, the padlock icon may pop up instead of the OSD menu, but the brightness won't increase anymore.
4. To eliminate the possibility of this issue relating to your computer, unplug the monitor from your computer and see if the OSD menu continues to "pop up". It if continues, you might want to consider purchasing a new screen.