Taipei and Hong Kong

The flight is so incredibly long that you should buy the best seat you can afford!

If you haven't yet booked your flights you might consider a multi-city flight plan: San Francisco to Taipei, Taipei to Hong Kong, Hong Kong to Beijing or Shanghai, then back to San Francisco. Would save you both time and money by not having to backtrack.
You'll need a visa to enter mainland China, though one is not required for Hong Kong. There's a simplified procedure at the airport in Hong Kong where they'll process a Chinese visa for you on the spot. Takes a few hours and costs more than doing it in advance from the USA, but it is very convenient.
One possible side trip from Hong Kong is the ferry over to Macau - fascinating place with a lot of history from its days as a Portugese colony. Can also book trips across the border to the southern Chinese provinces. I frankly forget whether you need a visa for that . Hong Kong is a Special Administrative Region of China, ie special rules apply which is why it's almost treated as a separate country when border crossings come into play...hence the requirement for the visa.
If you don't want to mess with all that you could do a quick round trip to Seoul or Tokyo from Taipei.
IMO, Hong Kong is a fascinating travel destination in its own right...worth more than 3 days anyway.

I have been to Taipei at least a half dozen times on business, and have had the chance to do some tourist stuff there as well. I did not find it to be a city of big touristic interest, but walking around the hectic urban environment can be very interesting. Going to the top of Taipei 101 is worthwhile. Watch out for scooters on the streets of Taipei! I think three full days in Taipei would be plenty, but I have always heard that the island outside of Taipei is very beautiful so you should definitely explore options beyond the city. The climate is warm and humid, and I would think in July could be even more so. So be sure and book a hotel with good air conditioning. Look into overnight flights—I used to take the midnight flight out of SFO, and that was easy. This was on EVA Air, and maybe they have a late flight out of LAX too. When you get on the plane at midnight, they serve a quick meal (which I never took part in) and then everyone settles down and goes to sleep. Passengers on that route seem to be well-versed on how it works, whether traveling on business or leisure. If there is any flight you can sleep on, it's one that leaves at midnight.

Hong Kong has lots to see and do, I'd just spend the extra time there. You can visit Macau as a day trip or spend a night or two. Macau is very different from Hong Kong.

If you're at all interested in a side trip to Tokyo, know that there's an alternative to Narita - the international airport located about an hour and 10 minutes from the city center. The domestic airport is called Haneda (HND) and is located only about 25 minutes from the city - with much more convenient (and cheaper) access to the center of Tokyo.
Believe that Tiger Air Taiwan has direct flights from Taipei into Haneda.

Taiwan is such a beautiful country and it's not all about Taipei. You should also consider visiting Kaohsiung, a little on the country side. But if you do really fancy going somewhere outside of Taiwan, you may consider going to Macau but 2 days is enough for Macau :)

Happy travels!