The stock market is likely to double during the next 10 years and then double again in the 10 years after that, as the economy resumes its historical expansion path, says Ron Baron, CEO of Baron Capital.

"The most important thing to think about right now is that growth of our economy is increasing," he tells CNBC. "Housing is doing great. Cars are doing great. Energy costs are low. Interest costs are low. Credit is widely available. Deficits are falling."

The economy grew 3.2 percent in the fourth quarter, although that number is expected to be revised downward.

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As for the stock market, "stock prices are valued at the median level they have been for 50 or 100 years, . . . or 15 1/2 times [earnings]," Baron notes. "It doesn't feel expensive to me with the economy improving the way it is."

Nominal economic growth, not adjusted for inflation, has averaged 6.6 percent a year since 1960, he adds. And he sees the stock market matching the economy's growth — "almost 7 percent a year, for a very long period of time, 10 or 20 years."

A 7 percent annual gain means the market would double in 10 years and then again in 20 years. "So that means 30,000 for the market [Dow Jones Industrial Average] 10 years from now and 60,000 in 20 years," Baron explains.

"It might sound like a huge number, but that's what compounding is. It makes small numbers get to be real big. In 1982, the market was 880. And now it's 16,000.

"The most important thing is that the stock market is a hedge against inflation," he stresses.

He said he personally invests in his company's mutual funds every month, sometimes every week — "what most people would regard as a substantial amount."

And why is that? "I think over the long term, the stock market will be much higher than it is right now," Baron states. "I can't tell you what it will be tomorrow, next week or next month. But I think the long term will be much higher, as it has been in my life."

Many other experts are bullish on stocks too. The Standard & Poor's 500 Index reached a new record high Monday amid belief the economy will thrive despite the Fed's tapering of its bond buying.

"U.S. equities can go higher in 2014," Drew Wilson, an investment analyst with Fenimore Asset Management, tells Bloomberg. "I believe the taper is not going to affect the real economy. I don't know what it'll do psychologically. I believe the fundamental recovery is real and will be strong enough to overcome the psychology."

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