The GEOMETRY of the visible crescent depends on the elongation (= angular distance between the center of the sun and the center of the moon). The elongation at conjunction (=new moon) varies between 0 degrees (at a solar eclipse) and more than 5.1 degrees. (This variation is a result of the tilt of the moon orbit against the earth orbit.) There should be a thin crescent at almost any elongation, though that crescent gets extremely thin and faint and also shortens at small elongations.
The elongation also depends on your location on earth, which can add up to one degree to the value calculated for the geometrical center of the earth. That is the difference between "geo-centric" and "topo-centric" elongation.
The VISIBILITY of the crescent depends on a multitude of factors, which can NOT be reduced to a single criterion, such as the "time since the conjunction". The greater the elongation, the easier it should be to see the crescent, BUT other factors such as the height above the horizon and foremost the external conditions such as the transparency of the atmosphere and the general weather play a DECISIVE role. Extreme crescents are probably invisible from sea-level.
An important question is, whether the sun must be below the horizont, or whether a sighting in broad daylight is usefull. Not sure, if that is realistic, though. Daylight observation works for imaging. Of course, if the sun is visible, extreme caution must be taken, not to get blinded by the light of the bright sun entering the telescope.
Ideal circumstances would be in earth orbit, on a specifically designed space station. From such a plattform, with a properly designed telescope, a crescent should be visible at almost any elongation, or at least the disk of the moon in earth-shine. You should be able to buy such an opportunity for some 50 million euros. ;-)More detailed discussion