Expanding analog inputs to the Arduino
There are two methods of doing what you want:


Add more ADC channels
Multiplex the existing ADC channels


SPI or I2C ADC chips are readily available in a range of resolutions, sampling speeds and number of channels.  They are fairly simple to add to any Arduino.

For instance the MCP3208 will give 8 channels of 12-bit resolution on SPI, which means 3 pins (MOSI/MISO/SCK) + 1 per chip (SS). So 1 chip would be 4 pins, 2 chips 5 pins, 3 chips 6 pins, etc.

Adding lots of chips to the SPI bus though can itself be troublesome with the increased capacitance of all those inputs meaning you have to reduce your communication speed somewhat or add extra buffering to drive the bus with more current.

I2C chips can be harder to have lots of them since there are only a limited number of addresses on an I2C bus - plus on many Arduinos the I2C is also two of the analog pins, which you may not want to sacrifice. 

The second option involves using analog multiplexers (eg the 4051) to switch different sources in to the existing analog inputs.

A third option which you probably haven't considered is to have multiple arduinos (or other low-cost microcontrollers) each doing some of the sampling and then implementing some kind of communication method between them (or to a single master). This has the added advantage that it's then possible to sample multiple channels at once (one per microcontroller) speeding up your operation somewhat.