Yes, you can buy exactly one share of a stock. In fact, many modern brokers allow you to buy even less than a full share through "fractional shares," meaning you can start investing with as little as $1.
Minimum Share Purchase Guidelines: How to Start Investing in Equity can you buy one stock
To buy a stock, you must open an account with a licensed stockbroker who executes trades on your behalf. Yes, you can buy exactly one share of a stock