The problem is very subtle because the reasoning is 100% correct. So where is the fallacy?
Time is Money
The issue comes when people equate a company's present value with its present book value. When a business grows, its net assets (e.g. cash, inventory) increases at a rate greater than its liabilities (e.g. debt). The difference between assets and liabilities is the company's equity or book value (this is known as the accounting equation).
However, the value of a company also incorporates future growth. If a company has a book value of $1 million and has a steady income of $100,000 per year, it's present value is certainly going to be greater than $1 million. This is just as if I had $100 of cash that doubled each year, I certainly wouldn't sell it for $100.
So it is true that if the value increases, the stock price increases. But the present value of a company already takes future growth into account. By buying the stock of a growing business, you are betting your money on the business growing faster than current market expectations, not on the fact that it will grow at all.
An investment strategy based around these "above expected" growth rates is known as growth investing. Growth related strategies generally target smaller, up and coming businesses whose potential isn't well known or analyzed. Internet and technology stock typically fall under this category. The same goes with emerging markets (e.g. foreign companies in growing countries like China and India).
Growth investing isn't necessarily bad, but it is misunderstood by many people. I find that in particular, people get overly excited about technology stocks. They see an excellent product, a rapidly growing user base, and a lot of hype surrounding a tech company. But then they fail to realize that other investors have already taken that into account and bought the stock up to it's current price. That is to say, the current stock price may already reflect everything that you considered.