I found this problem very interesting.
Maybe this sounds like a no-brainer, but here's how you can do an approximation:
Create a circle with a fixed diameter, say 1 mm. Duplicate it and make a blend from the two. Fit this blend to the curve from the start to end points and and increase the number of steps until the edges of the circles just touch each other and runs through the entire length of the curve. Go to the wiremode view to view this and zoom as close as you can so you could increase the number of steps and see when the edges just touch each other without gaps or overlaps. See how many circles are needed and you have a pretty good approximation. You may have to cut up the curve into sections if it is too long. The smaller the diameter of the circles in the blend, the more accurate the measurement will be. Tried it on a circle 100 mm in diameter and managed to fit 312 1 mm circles along the circumference. Remember that the number of steps does not inclue the start and end objects, so there are actually 314 circles running around it. Quite a good approximation. For open ended curves, you'd have to take into account the start and end portions of the circles extending beyond the line and subtract 1 circle from the total number of circles along the curve.