The obvious answer is no one knows.
However, one can in general, agree that if Company A grows its earnings, this will be reflected in a higher price of its stock. Therefore to answer the question objectively, one would need to know exactly what will be company A's earnings for next year and in 5 years time. As this is impossible, one can only guess or have a good 'hunch' about its future earnings. Accordingly, there will always a speculative element in your estimates of future earnings when you invest in a stock.
Nevertheless, price of a stock is intimately linked to earnings. When the company increases its earnings, this will be reflected in its share price also increasing. However, short term volatility in price can be large and the price correlation with the earnings likewise volatile over the short term. It is important for long term investors to know that the correlation between price and earnings over the long term is indeed very strong. This relationship is to be exploited by the intelligent investors.
Therefore, when asked if the share price of Company A will double in 5 years from today, assuming that its present price is fair price, an appropriate answer might be be, definitely yes, IF it can double its earnings in 5 years.
For earnings to double in 5 years, the EARNINGS GROWTH RATE should be about 15% per year over these 5 years. The company growing its earnings at lower than 15% per year is less likely to double its share price in 5 years from its fair price. (Its share price may double at lower earnings growth rate if it started off severely undervalued.) For example, a company with earnings growing at 7% per year is anticipated to see its share price doubled in 10 years.
On the other hand, a company growing its earnings at greater than 15% per year sustainably, will see its share price doubling in 5 years. However, a company growing at high growth rates carries with it certain risks related to this fast growth. Another paradox of a high earnings growth company is that its share price tends to be high due to popularity of the company amongst investors. Therefore, though it may be a great company, it may not be a great investment if bought at very high price.