The stock price of the East India Company jumped all over the place over its 200 year history, but it rarely went below the original issuing price of 100 pounds sterling per share.

The EIC became a permanent joint-stock corporation in 1657. Prior to that, each voyage to the Indies had been organized as its own corporation, lasting for the duration of the voyage, and being dissolved after the voyage ended. From the 1650s through the 1690s, the price of corporation endured a long decline, especially in the financial speculation that followed after the Glorious Revolution. The price bottomed out at 39 pounds sterling per share in 1698, when the crown granted a second royal charter for a competitor to the East India Company. However, the competition was short-lived, and the East India Company soon regained its monopoly on trade, and by 1700, its share price was back up to 100 and climbing, reaching a price of 200 pounds by 1717.

Then, like all investments in Britain, the East India Company's shares got caught up in the speculation surrounding the South Sea Bubble. The price skyrocketed, going from 200 pounds to 420 pounds by 1720. When the bubble popped, prices cratered, and the stock sold for 150 pounds in the summer of 1721. As the market stabilized, the share price grew steadily but gradually until 1757.

In 1757, Robert Clive won the Battle of Plassey, essentially granting the East India Company ownership over large parts of Bengal. Notably, this ownership came with the right to collect taxes. As a result, the share price surged once again, reaching a peak of 276 pounds in 1769. From 1769, however, a combination of parliamentary inquiries into corruption at home combined with political reverses in India put downward pressure on the stock price. (As a side note, one of the causes of the American Revolution was the British government's attempts to prop up the stock price of the EIC by granting it a monopoly over the American tea market.) The stock price declined for 15 years, bottoming out at 122 pounds in 1784.

In 1784, Parliament pushed through a set of reforms that stabilized the company's finances and granted it a government backstop in exchange for the company handing over political control over its holdings in India to a 5-man board of control established by the king. Though the company kept the lucrative diwani that enabled it collect taxes, this marked the beginning of the end of the company as an entity distinct from the British state.

With the end of its control over Bengal, the company started to look further east for revenue. It got into the opium trade, eventually becoming the primary supplier of opium to China. While this was terrible for the Chinese government, and led to two wars with Britain, the trade was very good for the stock price of the company, which would rise to a third peak of 298 pounds sterling in 1824.

However, by this point the government of Britain had become increasingly impatient with the EIC's troublemaking in Asia, and began to more rapidly erode the Company's corporate sovereignty. In 1833, the East India Company was essentially turned into an annuity by Parliament, who amended the EIC's corporate charter to strip it of its corporate interests and pay further dividends out of the revenues collected from the Indian diwani. The Company was finally rolled up 1874 when the government bought out the remaining shareholders of the EIC for 200 pounds per share.

So to answer your question, you would have done all right if you'd bought in in the 1650s, 1750s and 1850s. Those were all periods when the stock prices were all on the upswing from a trough. You probably would have done the best if you'd bought in during the early 1700s, though, and sold right before the South Sea Bubble cratered.

tl;dr: Here's a chart of the EIC's share price over its entire history. Note the giant spike that's the South Sea Bubble, and the smaller but more sustained spike right after the Battle of Plassey.

Source: Nick Robins The Corporation That Changed The World

EDIT: changed source link from Amazon to WorldCat after reviewing subreddit rules.