Have you heard of "mad cow" disease? This disease can come from eating beef or sheep that have been infected with an unusual infectious agent called a "prion". Prions are an odd form of protein and contain no nucleic acid (DNA or RNA), yet they are still infectious.

So how do prions reproduce without having DNA to provide instructions? Prions reproduce by converting certain cellular proteins into an infectious form. You might say that it makes the cell infect itself (and other cells that get exposed from a person eating prion-contaminated meat). The infectious prion is a changed structural form, dominated by conversion of protein helix structure into flat sheets. The infectious prion is extremely insoluble and resistant to protein-digesting enzymes, heat or other denaturing agents.

The prions attack nervous tissue, especially the brain. That is why the disease is called "mad cow". Affected cows run around uncontrollably and act crazy. When you examine the brain of such cows, the brain is full of holes, like a sponge. That is why they call it a Spongiform Encephalopathy ("enceph" refers to the head and "pathy" refers to "diseased").

More detail can be found at:
http://www.lmb.uni-muenchen.de/groups/winnacker/weiss/research.htm