David Wilson writes:

<<
I thought it was kind of neat that Borromean rings cannot be formed using
circular rings (of equal or differing sizes).

A related question: Given three rigid rings which are not all circular and are
reasonably smooth locally, is it always possible to form Borromean rings?
I was inclined to think this was true, but then I thought it might not be
possible if the rings were sufficiently tangled up that any attempt to form
Borromean rings would result in a more complicated linkage.

Is this question easier if we require the rings to be planar?
>>

Matthew Cook has a Borromean rings page:
<http://www.paradise.caltech.edu/~cook/Workshop/Math/Borromean/Borrring.html>
in which he presents several proofs of the impossibility of making the rings from perfect planar circles of any radii, and also makes the same conjecture (without any  smoothness condition).

--Dan