What is the maximum number of planes a controller can control?
There is no fix maximum a controller is allowed to have under control, as the other answer already pointed out. The airspace under control of ATC is divided up into sectors containing the airways to and from airports within that sector or in upper sectors, airways just for en-route traffic. (see also: Why do pilots use airways instead of GPS direct?).

Sectors can be merged during off-peak hours, split back into separate sectors during peak hours and in the case of TMA or approach sectors, the controller owning the sector can receive additional help by splitting the approach sector further and using different controllers for different tasks within that sector: Feeder/Pickup - Director - Departure Controller. The thresholds of when a sector is split is dependent on many factors, like weather, runway in use, outside influences like adjacent sectors having more traffic that usual, special occurances within that sector like parachute jumper drops. 

Sectors also vary in size and capacity based on their intended use. While TMA/Approach sectors are rather small and need to accomodate arriving traffic into an airfield, an enroute or upper sector can be significantly larger in size, lateral and vertical. Again, this is individual to each sector, because some factors which hold true for TMA sectors do not apply to enroute upper sectors and vice versa. It also really depends on the controller working the sector and his personal limits, favorite procedures and work practice. A few examples:


A TMA controller using vectors to the approach instead of standard arrival routes or transitions will have more workload to monitor and sequence aircraft.
The minimum radar separation in TMA sectors and upper sectors can be either 3NM or 5NM. While upper sectors are larger in size, they have to maintain a greater separation, so in some cases, an upper sector twice or multiple times the size of a TMA sector will only be able to accomodate the same number of aircraft (ceteris paribus) before reaching saturation.
A controller having to work with special situations such as aircraft in distress or student pilots requiring more attention than experienced pilots, will reduce the controllers personal capacity.


In summary, to answer your question: There is a situation and controller based threshold, at which saturation will be reached. Controllers are trained to anticipate this threshold and make their supervisor aware of the situation beforehand, so that measures can be taken. There is however no fixed numerical maximum, unless specified in the ARTCC/ACC's operating procedures.