The short answer is, no, you can't just stack them, but yes, it is possible (though not trivial) to connect ten or more of these to a single Arduino board and have it monitor all of them.
Note that the following explanation is simplified, possibly over-simplified, in the interests of brevity and giving you a general idea of what you're in for if you try this. Many of the details (as I mention below) could easily be one or several questions of their own.
You didn't give the datasheet for the board you're considering, so I'll work from the description of a similar board (though not an Arduino shield) using the same chip, the Cytron Technologies TC35 GSM Development Board.
Arduino shields connect various pins of their chips to pins on the Arduino board. In many (perhaps most) cases you can't have two different devices connected to the same Arduino pin, so if you were just going to plop another board on it would have to have special support for changing the pins it's using (often via jumpers) or being able to have several boards communicating on the same pins. The GSM chip you're considering uses two dedicated pins for serial communication transmit and receive; these are the pins you'll need to rewire if you're going to use multiple boards on a single Arduino.
What kind of rewiring you'll need to do depends on how many you really want. The Arduino Uno has 14 digital I/O pins, two of which are usually used for the USB serial interface to the computer, giving you six devices if you dedicate a pair of pins to each device. Since the communication rate with the boards is quite slow (9600 bps, see below) you could multiplex many of these on one pair of Arduino pins via a number of different techniques (such as using shift registers); this alone is complex enough that a different question could be devoted just to considering the possible choices, with further questions on the details of implementing each choice. You should consider, if you're interfacing with a PC, using multiple Arduino microcontrollers each with only one or a few GSM chips talking to them, talking to a PC via USB, and doing further processing on the PC.
If you're using a lot of boards you'll need to make sure you can supply enough power to them; you may need to use a power source outside the Arduino board itself. An Arduino powered from its USB interface can pull no more than 500 milliamps (because that's the USB limit). If you're using a separate power supply and the Arduino's on-board regulator, the limitation will be based on the electrical and thermal limitations of that regulator; this blog post is one detailed discussion of that.
There's also the issue of level shifting; Arduino boards usually do digital I/O at 5 volts or 3.3 V; this chip looks like it talks at 2.65 V and you or the board you're using will need to handle that. If the board you're using doesn't do appropriate level conversion there are other questions (such as this) that discuss the details of particular level-conversion solutions.
Once you have pins connected to the devices you'll need to talk to them; this involves the lower level "framing" parts of the serial protocol and the higher-level command set.
For framing the GSM chip appears to use the low-speed serial protocol framing that's standard in the computer world; aside from the voltage level difference it's the same as used in RS-232 communications between a computer and a modem or other "serial" device. The setting from the data sheet I used is "9600bps, 8-1-N, No Flow Control" which is slow enough that the microcontroller itself can easily communicate with quite a number of these without any extra hardware using bit banging. A 16 MHz Arduino executes something approaching 16 million instructions per second (most instructions are one clock cycle), giving you over 1500 instructions per bit-time to read a bit from every attached device, do the processing to turn the bits into bytes, and process the resulting messages. If you're careful with your programming, and depending on what you're doing, it shouldn't be too hard to deal with even up to a couple of dozen GSM chips, though this is almost certainly not a trivial programming job that you're going to do with standard Arduino libraries.
The higher level command set is also pretty much the same as old-fashioned computer-modem comunication; the GSM chip (and thus pretty much any board that uses it) talks standard PC-style serial protocol at 9600 bps using the AT command set. You'll see messages such as
+CMGR: "REC READ","+60125651453",,"12/07/31,10:0702+32"
when you receive an SMS and similar ones when you send an SMS.