Even without a prescription you can find most medications at the drugstore, even some that require prescription in other countries like asthma inhalers. However, for stronger pain killers you still need a prescription. You will find drugs cheaper in Taiwan than the West.
Drug Stores
In Taiwan, products may be different to the ones you usually take so it is always good to carry some from home, at least until you get used to your new environment.
Check what kind of drugs you can import without problem (and the permitted amounts) and ask your doctor for prescriptions as well, especially if you have medical needs or if some of the drugs are not completely allowed (they may pass the border if there is a medical proof for it, i.e., your doctor’s prescription). Sometimes when you visit a pharmacy, you will be given a strip of sealed plastic containing the daily dose of your medication. You just tear off the section for that day and take the pills as instructed.
Make a list of the name of the drugs in different countries and the basic compound so that it is easier to find them when you are in Taiwan.
Also, remember that, if you are not working, you will need to wait for four months in order to get your NHI card and so, it may be difficult (or expensive) to acquire some drugs.
Here is a list of drugs you can bring in small quantities:
As for eye products such as contact lenses, there is no need to bring them from home. They are widely available, cheap and very good quality.
Regarding deodorant and sun block, you can find it easily but it is expensive. As for cosmetics, they are widely available but if you have a skin condition or you prefer some brand in particular, you should bring your own.