DISCLAIMER: In no way am I suggesting that anyone lie on application for residency or abuse residency status for illegal activities. The description below is informative only of what has been done before by others.
If you have been living and working in Taiwan for five years, you could get an Article 51, Section 3 work permit and then corresponding ARC. You can work for any employer under this rule, bing long stands included, just as long as they have a registered company. You could find a Taiwanese/foreign friend who owns a Taiwan company to apply for you. Have the company pay your tax (i.e. you pay the tax) on the salary the company must show to the government. You could have all day to yourself, depending on whether the job actually means work or not.
Find a part-time job that gives an ARC. They are rare, but available. It would give you a half-day to do your own thing. Several goverment agencies have such jobs posted from time to time.
Go to school like the Shitda MTC. You give up two hours a day of freedom for an ARC. You qualify for it after four months of study. Do what you want otherwise. Come in on a student visitor visa which can be extended twice up to a total of six months. At month four transfer over to an ARC without leaving Taiwan. Many students I attended school with in the evening class (6:30-8:30) were doing just that. They had their own company (even one they owned), but the company wasn’t large enough to applly for work permits, so they just went to night school at Shita to keep residency.
BEST OPTION if you don’t need medical care or the need to apply for things that only an ARC can get you (like a phone account): Get a multiple re-entry visitor visa. I got one of these babys at the LA TECO office last fall. It is good for one year, meaning you can enter as many times as you like up to two month each time … and I hear that you can even go to the police station and extend each entry up to several months (confirm this). They gave it to me because I asked for it. I showed no documents explaining why I needed it. However, many TECO offices require you have some business-purpsoe to get this, like a letter from a Taiwan company or even a US-based company that shows you have a need/purpose to travel to Taiwan frequently. One may need a friend’s company to do this for you.
ADVICE: Whatever you do, get it in writing before you get on a plane and commit to some kind of visa run. I’ve saved mine arse several times, even when I didn’t follow procedure correctly, simply because I had a name, phone number and small chicken-scratch note of paper that had someone’s approval of a bogus process reported from some xiaojie in Taipei. The foreign TECO offices never want to double-cross people at the main Wai Jiao Bu in Taipei. So drop names when you have to and ask them to call Taipei if you run into trouble. They won’t call Taipei and will often lecture you on how you screwed up and how they are being nice and have a nice day Mr. XXXX.[/b]