it is a begonia, not sure which one though and wouldn't be hardy in the garden. You would need to lift it in the autumn and keep frost free during the winter.

has it ever flowered for you as they do. Perhaps more water and food to perk it up a bit?

Thank you Seaburngirl. Yes, it has flowered from time to time and in fact has a couple of delicate white flowers now, though they are not visible in my photo. What amazes me is that it is still in the same pot after all this time and has kept going. Maybe I'll move it to a larger one and put it somewhere else to see how it fares. Not too keen on repeated plantings and liftings - not as young as I was! (I've reached a stage in life where I am seriously considering hiring a gardener.)

it is likely to be a tuber/corm one which is why it still fits the pot. I'd treat it to a bigger pot and a goof feed if it were me.

Actually, I think it is one of the rhizomatous hybrids--maybe 'Kit Kat'--which also like to be crowded. This one looks like it is ready for either repotting or division, though.