Given a fixed area on a solar panel, at what point does adding cells in parallel not increase voltage?
If you are referring to a solar panel rather than a complete installation then doubling the number of cells would indeed get twice the amount of voltage but each cell would only be able to provide half the current so this will not increase the total amount of power available from the panel.
Since there is lost space at the edge of each cell you won't be able to get quite the same amount of area of cells so the power will be slightly less and it will cost more because of increased number of connections and manufacturing steps.
The voltage outputs of photovoltaic panels have traditionally standardized on a few voltages, in particular, voltage suitable for use with 12v lead-acid batteries so most arrays are designed around that. This will require about 32-36 cells in each panel to provide that voltage. The open-circuit voltage will typically be in the range of 18-21 volts with a maximum power voltage of ~12-15v - the same as a lead-acid battery.
For an entire array made up of many individual panels they are typically connected in series to give an operating voltage that can be up to several hundred volts. Beyond that the expense and difficulty of making the system safe creates a limit.