Strange loops is a term introduced by Douglas Hofstadter in his seminal book, G"odel, Escher, Bach: An Eternal Golden Braid. Simply put, it refers to self-referential (recursive) constructs. For a more detailed explanation, read the book; hopefully the examples below will illustrate my thinking on strange loops. This is in progress.
GEB ... likens inanimate molecules to meaningless symbol, and further likens selves (or "I"'s or "souls", if you prefer---whatever it is that distinugishes animate matter from inanimate matter) to certain special swirly, twisty, vortex-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols. It is these strange, twisty patterns that the book spends so much time on, because they are little known, little appreciated, counterintiutive, and quite filled with mystery. And for reasons that should not be too difficult to fathom, I call such strange, loopy patterns "strange loops"...
... the Gödelian strange loop that arises in formal systems in mathematics ... is a loop that allows such a system to "perceive itself", to talk about itself, to become "self-aware", and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self. ---Douglas Hofstadter, G"odel, Escher, Bach: An Eternal Golden Braid
The greatest strange loop in biology is the one where DNA is used to make proteins which in turn is used to make more DNA. In general self-replication of a system can be characterised by tangled hierarchies or Strange Loops and emergent/complex behaviour when a network is created between the individual components of the system within the confines of a selective environment.
One cool strange loop has to do with cellular influence on protein folding. Proteins (in the environment) require proteins (in the environment) to fold up. For example, chaperonines are proteins that help fold up other proteins.
Is there a chaperonin that requires another chaperonin to fold? Now that would be a beautiful meta strange loop!
One of the strange loops involves the independent/early folding unit (IFU) stabilisation, where these IFUs flicker in and out of an ensemble of conformations and in order to form a stable structure need to interact with each other. Likewise for larger sub-structures.
Beta-strand pairing can also be thought of as a strange loop, much like DNA base pairing is a strange loop. Beta-strand formation is similar to IFU formation in that the hydrogen bonding is required for overall stability.
If a chain folds upon itself sequentially starting from end to end (which we know is not the case) then the entire folding process would be a beautifully recursive strange loop, where to fold a protein of length n, you would fold a protein of length n-1 and then fold the last residue.
All of the above can be generalised. Strange loopiness arises because of context-sensitivity in proteins (as illustrated by the IFU and beta-strand pairing example). So to fold a protein of length n:
f(n) = f(n-1) + c(R_n) f(n-1) = f(n-1-1) + c(R_n-1) = f(n-2) + c(R_n-1) f(n-2) = f(n-2-1) + c(R_n-2) = f(n-3) + c(R_n-2) . . . f(3) = f(3-1) + c(R_3) f(2) = f(2-1) + c(R_2) f(1) = c(R_1)
where c(R_1) is some arbitrary starting point. I use methods like this to build up protein structures---in fact such a process was used in my CASP3 ab initio prediction method for assigning secondary structure to rough models. The concept is known as recursion.
Note that the recursive call need not be made to a sequnece of length n-1. It could be of any size that represents a substructure. So you can view the process as putting together a bunch of independent sub-units.