Once at 0% in Ireland and once at 15% for capital gains. Where can I sign up to have my income double taxed like this?

My understanding is that this is only possible for very profitable corporations. Can smaller entities get away with this?

You are taxed only once: with income tax. Company is also taxed only once: with corporate profit tax. The reason why that makes sense is because corporation is treated a separate legal entity (person), and that's a good thing for you, because you are not responsible for company's mistakes (that's why they are called "limited liability companies". Publicly traded companies also have limited liability).

Also, if, as a majority shareholder you appoint yourself as the CEO and pay yourself a salary, it is tax deductible for your company, so money that goes to your salary is taxed only once.

That's a reasonable first thought.

When I hear "double taxation" in the US I usually think first of self-employed people sometimes having to pay both income and payroll tax compared to an otherwise employed person only paying income tax.

Then maybe I think about places like Missouri where after you take your post-tax income and buy a car with sales tax on the purchase, you pay yearly for that car as a personal property tax (which does not include your license plate, or inspection, or the city sticker some local places require).