A circular definition occurred in an early definition of the kilogram. The kilogram was originally defined as the mass of one liter of water at standard pressure and the temperature at which it is densest (which is about 4 °C). The unit of pressure is the newton per square meter, where a newton is the force that accelerates one kilogram one meter per second squared. Thus the kilogram was defined in terms of itself. Since water is nearly incompressible, this circularity is of no consequence — with each iteration of the "circle," the resulting measure of a kilogram rapidly converges. Even so, to clear up any confusion, the kilogram was later defined as the mass of a certain piece of metal in Sèvres.
A circular definition also crept into the classic definition of death that was once "the permanent cessation of the flow of vital bodily fluids", which raised the question "what makes a fluid vital?"
A branch of mathematics called non-well-founded set theory allows for the construction of circular sets. Circular sets are good for modelling cycles and, despite the field's name, this area of mathematics is well founded. Computer science allows for procedures to be defined by using recursion. Such definitions are not circular as long as they terminate.
Another, commonly cited example:
Another one, not so common:
Once you find "Loop, endless"
The 2007 Webster dictionary defines a "hill" and a "mountain" this way: