A set of Brunnian link is a nontrivial link such that if one component is removed, it becomes trivial. The best known example is the Borromean rings:

Here's a six component example:

There is likely a brunnian link with infinitely many components by generalizing the previous example.

My question is, is there a brunnian link with infinitely many components, such that each component has less $n$ crossings with other components, for some finite $n$? (By crossings, I mean crossings in some fixed link diagram of the link).