RnD on an EEG, Help required, what op-amp?
Your average instrumentation amp can easily do 1MHz bandwidth; and your EEG should be no more than 2kSPS. So a multiplexer / Sample and Hold ahead of the the instrumentation amp ought to save you there.

But consider that the amp should be only a few dollars.  Is it worth the multiplexing?  If you do Surface Mount, the size will be quite minimal.

The Arduino cannot digitise faster than about 10kSPS, so you would need a faster A/D to do 16 channels.  Something that can do 12 bits at 100kSPS would be nice.  They are also fairly cheap.

Note that for patient safety you need optical isolation on signals, and a good isolated supply (battery or similar).  Don't mess with safety in this area - if you need to get a high speed data stream out, build your own isolators or use fibreoptics to transmit the signal.