Does anyone know if it’s possible to get Visa or MasterCards from the US without being a US resident, so I can buy stuff on iTunes?
Sure, it’s possible. There are several ways to do it but the easiest (and fastest) way is to get an address in the US, which can be done with one of the mail-forwarding / virtual office companies – USA2me.com and MyUSA.com are two of the most popular and reliable. It’s pretty cheap and it’s not a PO box. The main obstacle might be getting approved for the credit card itself. Contact your Taiwan bank to see if they have an affiliated US card or sponsor program. Otherwise, from what I’ve been told from my homeslice, credit cards are being floated by every major company right now, so it shouldn’t be a problem.
Don’t they have debit cards with the visa logo , or a prepaid “credit cards”?
Those mail forwarding companies are a good idea. I hadn’t thought of that. I use mom instead.
Anyway, that’s step 1: get a US address.
Then step 2: open a US bank account using that address. Might as well make it a checking account, so you’ll have the added convenience of being able to write USD checks, too.
Step 3: apply for a Visa card at that bank and link it to your checking account so the Visa bills are paid automatically from your account (otherwise, if you have the Visa bills sent to you in Taiwan you have to pay to wire $ to pay those bills and by the time you finally receive the bills here and get around to wiring the money you’re too late and interest was tagged on, so you didn’t pay it fully after all).
Step 4: deposit a large chunk of money into that checking account (a few thousand dollars), so you can use the Visa card freely without ever worrying about whether you have sufficient funds in the checking account to back it up.
Step 5: periodically replenish the funds in your checking account.
Electronic statements and payments. Welcome to the 21st century. 
Sorry I’m a little slow and I admit I’m only entering the 21st century, but are you suggesting making payments manually (electonically) every month? If so, that’s too much hassle for me. I just have them withdraw it automatically, so I don’t have to hassle with it (then check my statements afterwards to confirm all was proper).
You can also set it up to pay the minimum every month or to pay the full balance out of your bank account IIRC.
Huh? :eh: That’s what I do. My Visa is paid in full automatically out of my US checking account.
I was just talking about your reference to getting statements mailed to you and your reference to wiring money late. And you don’t need to get a VISA at the same bank as your checking account. If you’re using it in Taiwan you might as well get it at the same bank since it’s a cash economy here. But in the US I use a REI Visa card which gives 1% cash back (not exactly cash but rather camping gear) and that adds up when nearly everything except rent is paid for with a CC.