With countertop surfaces the thickness of the edge varies by the material. For the most part, the differences for the standard edge thicknesses would not be that noticeable to the human eye unless they were side by side.
With Solid Surface countertops (i.e. Dupont Corian(r) and LG Hi-Macs(r) ) the standard thickness is an inch and a half. The thickness of the material is only a half inch but the edge is built up to give it a much more natural thickness. That being said the thickness of a Corian(r) edge is the most customizable of any surface. Because of the nature of the material you can really have any thickness of edge you desire. Whether it is 1 inch, 2 inches 3 inches or more. The options are close to limitless when it comes to solid surface.
For stone based products like granite and quartz countertops (i.e. Dupont Zodiaq(r), Caesarstone(r), and LG Viatera(r) ) the standard thickness is either 2cm or 3cm. 2cm is equal to about 3/4" and 3 cm is just shy of 1 and a 1/4 inches. 2cm edges can be laminated to be double as thick. The same can be done with 3cm but it is not a common application.