Supposing I wanted to connect to the ISS's WiFi network from the ground, how large of an antenna would I need to do so, and what kind? Would they need to increase the power on their end or use a specialized antenna of their own? By how much?
Antenna design is about more than just size. In fact antenna elements themselves have maximum useful size (order of one wavelength). But the short story is you cant with any size. The long answer is of course there is always a way, but it would start to get very silly. WiFi frequencies and protocols just dont work over those distances.
Not exactly connecting to a regular Wi-Fi network, but shows what is possible using the suitable configuration. Dish antennas were used on both sides.
The longest unamplified Wi-Fi link is a 304 km link achieved by CISAR (Italian Center for Radio Activities).
Antenna is 120 cm with handmade waveguide. 35 dBi estimated
Ignoring the latency (actually not that bad) and faraday cage problems, the main problem here is the radio link budget.
The link budget is a calculation for how much usable bandwidth (data rate) can be maintained for a certain set of parameters. The parameters the receiver can optimize are the Receiver Antenna Gain and Receiver Losses. Let’s assume Receiver losses (from the receiver hardware) are as optimal as they can be. Now there’s only one parameter to optimize: Antenna Gain.
Antenna Gain results from two more parameters: antenna size and directivity. As said in the other answers, size provides significantly diminished returns above a single wavelength, which is on the order of ~10 centimeters for 2.4GHz WiFi. Now we are down to our last parameter, directivity.
Directivity is a result of the design and shape of the antenna, and it trades margin for error for extra link budget (to be spent on range or data rate). If the antenna can be pointed precisely in the right direction, this extra Gain can be used. For this extreme case, we would immediately go to the most directive antenna type, a parabolic dish (sometimes called a satellite dish). An interesting thing about this type of antenna is that it pretty much removes the size limit.
Wikipedia has some numbers to reference:
It can be seen that, as with any aperture antenna, the larger the aperture is, compared to the wavelength, the higher the gain. The gain increases with the square of the ratio of aperture width to wavelength, so large parabolic antennas, such as those used for spacecraft communication and radio telescopes, can have extremely high gain. Applying the above formula to the 25-meter-diameter antennas often used in radio telescope arrays and satellite ground antennas at a wavelength of 21 cm (1.42 GHz, a common radio astronomy frequency), yields an approximate maximum gain of 140,000 times or about 50 dBi (decibels above the isotropic level). The largest parabolic dish antennas in the world are the Five-hundred-meter Aperture Spherical radio Telescope in southwest China, and the Arecibo radio telescope in Arecibo, Puerto Rico, US, which both have effective apertures of about 300 meters. The gain of these dishes at 3 GHz is roughly 90 million, or 80 dBi.
If you really want to calculate this, try using the link budget equation with transmitter specs taken off a consumer WiFi access point. An estimate I have for the antenna size would be ~20 meters, resembling the antennas from the SETI radio astronomy arrays.
Another, harder-to-calculate way to improve Gain is to use an array of antennae, with complex hardware or software for combining the weak, noisy signals from each into a strong, clear one.