Sugar dissolution test
The sugar will keep dissolving until it reaches it's saturation concentration of around $\pu{900 g/L}$, which is of course horribly sweet, thick and undrinkable as a beverage.  Assuming you don't completely saturate the water with sugar in this way, the undissolved sugar will continue to dissolve so long as there is still any remaining in the glass.  

A common way of measuring the dissolved sugar concentration is by refractometry, as described below (note that home refractometers can be purchased for as little as $30 US):  


  A refractometer is a laboratory or field device for the measurement of
  an index of refraction (refractometry). The index of refraction is
  calculated from Snell's law while for mixtures, the index of
  refraction can be calculated from the composition of the material
  using several mixing rules such as the Gladstone–Dale relation and
  Lorentz–Lorenz equation.
  ...
  In homebrewing, a refractometer is
  used to measure the specific gravity before fermentation to determine
  the amount of fermentable sugars which will potentially be converted
  to alcohol.


So, in your case a "homebrewing" type of refractometer is probably your best bet. As long as sugar remains in solution, it will continue to dissolve until it reaches saturation.  This also means it's refractive index and specific gravity will continue to increase.  Once this increase is no longer observed, then the sugar is no longer dissolving, meaning you've either reached saturation or all of the sugar has dissolved.  Again, saturation would mean a thick, honey-like solution. Thorough stirring throughout the measurement process is a must.