I’m a Brit too. I got a sixty day non-extendable in HK, with very little fuss back in June. I remember giving them a printout (easily altered) of an online banking statement. Had to tell them I was a tourist.
Went back in August and got another 60 days, this time extendable. Needed proof of funds, letter of acceptance to study, a letter from me explaining why I wanted to stay in Taiwan(study), and a return ticket.
Met some South Africans on the way home who had no problems getting their second 60 days from the Bangkok office.
Due to changing jobs I needed another visa this month. Despite overstaying I got 60 days non-extendable in HK. The lady I dealt with was actually very helpful, although the dragon at reception is a different story. All I really needed was a letter from the education department stating that they were processing my application for a work permit.
HK is also good because you can fly there on a one-way ticket from the UK and still enter without a visa. If your job is for a year or more you won’t be able to use the return part of a round-ticket ticket anyway.
Then buy a return to Taiwan, which means you have the cheapest possible solution to the ‘ticket home’ requirement. I actually bought my ticket in China: Haikou-Hong Kong/Hong Kong-Taipei/Taipei-Hong Kong. I have kept the unused portion of my ticket, and bought a new return from Taiwan for each visa run. Nobody has ever questioned it.