A fairly well-documented article in Wikipedia, a "List of long-living organisms," explores different ways in which organisms can get old. A number of other clonal plants and animals are mentioned. Some clonal plants are estimated to be tens of thousands of years old, not because they leave layers of wood, but because they spread slowly outward, sprouting new roots and abandoning the original center of the colony to decay. They form circular colonies similar to a fairy ring of mushrooms (fungal colonies likewise are potentially quite old). A colony of quaking aspen (Populus tremuloides) - trees that propagate asexually from their root system, have an age estimated by some at over 80,000 years, but could be much older.
|Clonal plants that spread by horizontal
rhizomes, like this giant Gunnera, may
potentially be the oldest living "individuals,"
but their age cannot really be measured.
If we're talking about an integrated set of organs and tissues in a discrete individual, than a 120-year old human being is more impressive than any plant. Some of our tissues turn over during our lifetime, but some do not. I'm amazed that my own brain is still working at all! Some kinds of tortoises can live more than 170 years, and koi fish have been reported to be over 200 years old. Invertebrates like sea urchins and bivalve molluscs have been recorded at more than 200 years as well. (Sponges may live for 10,000 years or more, but are technically more like clonal colonies than individual animals.) Animals are different from plants, beginning small, but complete, and getting bigger with age. We have only one set of legs, eyes, etc., that have to last a lifetime. So we are really older than any tree or clonal organism.
So you can make an argument for many different "oldest" organisms, depending on how you set the rules.