Is there any independent non-DNA based information system in the cell
According to the comments, the question is about inheritance systems of cells. The "mainstream" information storage and inheritance system is of course DNA (and RNA in some cases). However, there are additional and parallel inheritance systems even within a single cell (not to mention in multicellular organisms). These form a subset of epigenetic inheritance methods and - as expected - are less precise than the information copying mechanisms of polynucleotide replication (in eukaryotes) and certainly more limited in the amount of information that can stably maintained throughout generations. This was explored in detail by Eva Jablonka and Marion Lamb (among others) in a few books and many papers:


Jablonka, E. &amp; Lamb, M. J. 1995. Epigenetic inheritance and evolution: The Lamarckian dimension. Oxford University Press, 346 pp.
Jablonka, E. &amp; Lamb, M. J. 2006. Evolution in four dimensions: Genetic, Epigenetic, Behavioral, and Symbolic Variation in the History of Life. MIT Press, 472 pp.
Jablonka, E. &amp; Szathmáry, E. 1995 The evolution of information storage and heredity. Trends in Ecology and Evolution, 10, pp. 206-211.
Maynard Smith, J. 1990 Models of a dual inheritance system. Journal
of Theoretical Biology, 143, p. 41-53.



  “An epigenetic ineritance system (EIS) is a system that enables a
  particular functional state or structural element to be transmitted
  from one cell generation to the next, even when the stimulus that
  originally induced it is no longer present. In other words, EISs are
  the systems that enable the transmission of various phenotypic
  expressions of the genetic information in an individual.” (Jablonka &amp; Lamb 1995, p. 80.)


Epigenetic inheritance methods include the following systems that could in theory code for information that is potentially inherited from parent to daughter. The list is far from being exhaustive.


DNA methylation: a methyl group added to a gene's promoter represses its transcription. Methylation is heritable: when DNA replicates, methylation carries over. There are other chromatin-marking methods that are inherited.
Steady-state inheritance system: "the inheritance of functional states through self-sustaining metabolic networks, maintained by positive feedback." (Jablonka &amp; Szathmáry 1995 TREE) The cytoplasm itself is also inherited by the daughter cells, but any information "stored" in the cytoplasm cannot effectively be "replicated" from cell to cell.
Structural inheritance system: 3D structures of the cell provide the template for similar structures to assemble, e.g. "transmission of cytoskeletal organization and of cortical organization" (Jablonka &amp; Szathmáry 1995 TREE).
Symbionts and parasites (bacteria, viruses, plasmids, etc.) that can replicate within the cell (so that when the host divides, both daughter lineages receive the partner). Note, however, that if the parasite is too virulent then it is expected that horizontal transfer is more emphasized than vertical "inheritance".
Prion proteins. Prions are proteins that may exist in several distinct conformations, and at least one of these conformations is capable of replication by forcing its alternate conformation (and therefore alternate function) on the normal physiological form of the protein (both having the same amino-acid sequence). See Shorter, J. &amp; Lindquist, S. 2005. Prions as adaptive conduits of memory and inheritance. Nature Reviews Genetics 6, p. 435–450.
...