In specific situations you can get away with having only one ADC by multiplexing your sensors. In my scenario, because I was measuring totally unrelated things, this was not possible.
The most obvious approach is to add an external ADC. The drawback is that this requires you to get to grips with how to control your particular ADC chip, and beyond that to get serial communications working correctly. A further negative is that these ADCs usually require additional components in the form of stabilisation capacitors.
I therefore came up with a simpler alternative – using an analog switch ic. I was pleased to discover that they are now available in extremely small form factors. Actually, my choice turned out to be rather too small – at a size of 1 mm x 2 mm, soldering directly to each pin (without the benefit of pcb mounting) proved more of a challenge than I was looking for.
More details at Tinkermax’s blog.