~ a flat sheet. They just don't use them and you just can't buy them. Don't bother bringing a whole set of sheets, as they won't fit, but you'll be happy to have a flat sheet.
~ cold medicine. You will get sick, and cold medicine technology in Taiwan is kind of stuck in the 80s. So bring your Nyquil and Aleve Cold & Sinus.
~ Reese's PB cups, York Peppermint Patties. If you like them, bring a bag. They're tough to find. Snickers, Twix, M&Ms and Doritos are all at your local 7-11.
~ an electric blanket if you like them and you're going to be there during the winter. I didn't have one but there were nights I wished I did.
For the ladies:
~ All the shoes you'll need, if you wear anything bigger than an 8. You can buy Birkenstocks and there are a few specialty shops for shoes but they're expensive. And bring comfortable shoes.
~ All the clothes you'll need, if you wear anything bigger than an 8, or are above maybe 5'6". I'm 5'8" and a 10 (US) and I just barely didn't fit into the biggest size in stores. There are big-ladies shops but tend to fit an Asian body, and are usually too short or just awkward-looking.
~ "feminine hygiene" these products are also stuck in the 80s. You'll be able to find good ones after you've shopped around but it's nice to have a supply when you first get here.
What to buy here:
~ A puffer jacket. When it's 50 degrees and drizzling you will want to be bundled up. These are easy to buy in any size.
~ Glasses. Relatively cheap and easy to get, and there are eyeglasses stores everywhere.
~ Prescription medicine. Bring a months' supply to tide you over, but even without health insurance, it's easy to get at pharmacies. You can't get everything at every pharmacy but ask around and you can find a place to buy everything from allergy medicine to birth control pills to Xanax over the counter, probably cheaper than you paid for a co-pay in the US.
~ vitamins, shampoo, soap, face wash, toothpaste, makeup.... Basic recognizable brands (Colgate, Dove, Pantene, Cover Girl) are all here and there's usually enough English that you know what you're getting. Department stores like Sogo have all the expensive brands and Body Shop. There's also a few brands I didn't know of in the US that are natural and sulphate-free, readily available at Carrefour and Watsons.