What determines the maximum speeds for a particular aircraft?
There are various physical things that will limit the top speed of an aircraft. Which one you will hit first depends on the airframe in question but here are some things you may encounter (aircraft dependent): 


Drag will generally counter your thrust which will limit the speed you can achieve in level flight and is often the practical limiting factor on aircraft that actually fly. Level flight is important to note as you can gain a huge amount of speed if you simply enter a dive. 
For propellor driven aircraft there are some practical speed limits because of the propellors.
At some point flutter will become a problem and you may literally shake the wings off the plane. 
Around the speed of sound (mach 1) you are going to have all sorts of interesting things going on if you are pushing a subsonic optimized airframe to such speeds.  
Ultimately, if you still have your engines and wings heat becomes the real problem. The Concorde pumped fuel through the nose to use a coolant in the mach flight regime and was in part speed limited by nose temperature. For manned flight this becomes the limiting factor at the extreme end of the envelope and aircraft like the SR-71 had a very physically demanding environment.