Do any enzymes exist that are made of something other than protein, DNA or RNA?

Enzymes are proteins, and are made up chains of amino acids like other proteins. However, there have been some other discoveries that cause us to deviate from that definition slightly.  Molecules called ribozymes have been found in nature that are small segments of RNA that can catalyze reactions and therefore act like enzymes. For example, the part of the ribosome that links amino acids together acts like a ribozyme. Because of their ability to carry genetic information and catalyze reactions, ribozymes are studied closely as a possible clue to the beginnings of self-replicating systems in early life. 

In nature, we do not yet know of any natural enzymes made up of DNA, but they have been made artificially in the lab, as is seen in the paper I've referenced below. These molecules are referred to as deoxyribozymes. Both ribozymes and deoxyribozymes are interesting to researchers for their possible applications in living systems as tools for research or even therapeutic agents. 

Many enzymes require cofactors or coenzymes in order to function properly. Cofactors and coenzymes can be inorganic, such as metal ions, or organic such as vitamins, or even derived from other proteins. These are not part of the chain(s) of amino acids that make up the enzyme, but they bind to the enzyme at a binding site. 

In short, protein, RNA and DNA covers the range of materials that enzymes can be made up of for now. I say, 'for now' because new discoveries are always being made so my explanation could be proven wrong in the future. Remember that other molecules can be used as cofactors and coenzymes, but they are not generally considered to be part of the enzyme itself.