Is it illegal to give away stock?

As noahc says, this question is too vague to be useful. But here's some examples of rules regarding stock transfer.

If you're in the US and talking about stock in a private company, stock is often "not transferable". You can choose to sell it, but you might only be allowed to sell it to current investors, and the company might have a clause that says that if you try to sell it to someone they don't want you to sell it to then they have the option to buy it instead of the other person, and there's all kind of things like rights of participation that are designed to keep ownership percentages in line, and it's very complicated.

Note that these examples are not illegal, they're just disallowed by the terms set by the company that you acquired shares in. When you acquire the shares, you agree to the terms.

Without more details, particularly, which country you're in no one can answer this question. Are you giving it to a person or a non-profit? How much? Why are you doing it? Are you trying to make a company have more shareholders? Are you trying to 'share your winnings' ?

One specific problem (amongst the many others pointed out in this thread) is stock transfer generally triggering a tax event. Specifically, gifting of stock requires, by IRS -amongst other reporting requirements- for you to: 1, calculate the value of the company; 2, determine the valuation of the stock; and 3, pay taxes on given year against that valuation.

In most startup contexts, 1 & 2 is both costly, time-consuming, and non-productive, while 3 is not really desirable. There are structured ways to have founders, and stakeholders stock (or first proxies thereof) of the company in their assets without triggering these. Kindly consult with your accountant to figure out what these might be.

of course not, you do it every month or so through proxy voting...

seriously though, wouldn't it be considered a gift? if so and my thinking is correct and gifts are taxable then... just be sure to pay your taxes

It depends on how you give it away.