Arguably, Rule 110 [1] fits the bill. Once you have your data, you just apply Rule 110 endlessly.
One could argue that one has merely embedded the actual operands into the data fed to Rule 110, and I would then observe that in the context of Turing completeness, there is a fundamental fuzziness between data and code that goes "all the way down". All your no-operands single-instruction can really be is "do it", with "it" specified by the data.
Right, this is very interesting, you are effectively running the data and code intermingled instead of in different separate memory areas.
A fascinating thing about computing with rule 110 seems to be that it is inherently parallel, the "instruction" is applied to the entire memory area in lockstep. There is no control flow of any kind. So if you have static data you need to encode it as "idle" instructions.
[edit: langton's ant suggests so - http://en.wikipedia.org/wiki/Langtons_ant]