You may scale the curve with Alt+S to make the objects grow gradually.
Add an object and parent it to the curve (Ctrl+P-->Object). In the Curve Data header check the Path Animation checkbox. Check the Follow box also if you like to fit the objects to the curve's shape.
In Object header (with your object selected) set the duplication type to Frames and uncheck the Speed checkbox.
Adjust the Frames number in Path Animation panel as you like. Scale your object and apply Rot&Scale to it (Ctrl+A).
In Edit Mode select one segment of the curve, snap the cursor to it, enable proportional editing and press Alt+S.
You can now modify the curve's shape without deforming the objects.
Using this method I've created the row of scales (rotated the scale a bit to make it look better). Disadvantage of this solution is the scaled objects are not spaced evenly.
NOTE: The other method, which uses inserted scales to different keyframes on a timeline (as @Leander suggested in his comment) works the same way.