There are currently no stock exchanges where you can trade company's stocks with bitcoin digital currency. Yet there are many exchanges where you can trade your dollars, euros and yuans for bitcoins.
For many investors, bitcoins started like a penny stock. It was quite easy to buy few hunder of them when bitcoins were prices at few pennies or bucks.
According to Bitcoin Charts, the biggest trade markets as in December 2013 are:
You should definitely check out Bitcoinity as they have real-time statistics about the volume and exchange rate of bitcoin in several exchanges and with various currencies.
Some exchanges started offering bitcoin derivates. One of the greatest bitcoin derivatives market is ICBIT. If you are into futures, you can find there more information about various currency and commodities contracts that you can trade with bitcoins.