Iceland boss Richard Walker said higher energy bills and other costs meant price rises were now "inevitable".
The warning came as analysts predicted that household energy bills could rise by hundreds of pounds next year.
They said the energy price cap, which protects domestic consumers, could soar by £400 in the spring.
Cornwall Insight forecasts that the energy price cap will rise to about £1,660 by next summer.
That is about 30% higher than the record £1,277 level for the cap set for winter 2021-22, which began at the start of October.
"With wholesale gas and electricity prices continuing to reach new records, successive supplier exits during September 2021 and a new level for the default tariff cap, the Great British energy market remains on edge for fresh volatility and further consolidation," said Craig Lowrey, senior consultant at Cornwall Insight.
Energy regulator Ofgem said the price cap "will ensure that consumers don't pay more than is absolutely necessary this winter".
But if gas prices stay high, the price cap will rise, Ofgem said.
The regulator said its "number one priority is to protect customers", but acknowledged "this is a worrying time for many people".
But while the price cap helps households, there is no such safeguard for businesses, which have to absorb the full impact of rising global energy prices.
Mr Walker warned that Iceland's energy bill would go up by £20m next year. Alongside higher salaries to address lorry driver shortages and other new costs, he said grocery prices would have to increase.
"It's inevitable that we will see price rises," he told the BBC. "The UK supermarket industry is one of the most competitive in the world.
"Our margins are very very tight and we're not an endless sponge that can just absorb all of these different cost increases."