In principle, you can charge a conductor indefinitely.
But remember that in order to cause a flow of charges from a body (call it a 'source') to another (the conductor in question), the potential of the former has to be lower than the potential of the latter.
This potential difference causes a current to flow from the source to the conductor, resulting in a transfer of charges.
All the charges that are already in the conductor will exert an electrostatic repulsion on the incoming ones, slowing them down and making it more and more difficult (as more and more charge accumulates in the conductor) to charge it further. As charge continues to flow out of the source and into the conductor, the potential difference decreases and the potentials of the two objects become more and more similar.
At some point, the potential difference will reach 0.
The current will stop flowing when the charge in the source is equal to the charge in the conductor, which corresponds to the situation in which the electrostatic repulsion from the charges in the conductor is equal to the force attempting to put more charges in. Here, the potential difference is 0.
If you want to force more charges to flow out of the source and into the conductor, therefore increasing its charge, you have two options:
1) you either increase the potential of the source, so as to create a non-zero potential difference and thus causing current to flow.
2) introduce a new force that pushes the charges out of the source and into the conductor: this force (eg chemical force) has the job of bringing a charge of the source against the electric field exerted by the charges of the conductor.
The real limit to the charging of a conductor would be when there is no physical space available for the electrons. Electrons are fermions and they cannot occupy the same position in space, so it will become harder and harder to squash them together.
NOW, in the context of capacitance:
Imagine a circuit with a generator (battery) and a capacitor.
DC current will flow only in half of the circuit (left or right depending on the choice of charge carries, electrons or ions, that is conventional).
Let's say protons carry the charge (this is the convention for current, although physically it is the electrons that to the moving).
Protons leave the + terminal and accumulate on the closest plate of the capacitor.
The plate is now the conductor, and the + terminal of the battery is the source.
The plate has initially 0 charge and therefore 0 potential.
The charges accumulating on the plate will exert an electric field that is going to oppose incoming protons. The potential of the + terminal decreases, the potential of the plate increases, the potential difference reaches 0 and current stops flowing: there are as many protons in the + terminal as there are on the plate.
Unless you increase the charge in the + terminal (therefore increasing its potential) or apply another force on the protons, no more current will flow.
If you now disconnect the battery and close the circuit, you have one of the capacitor plate charged (so at a non zero potential) and the other one with no charge ( 0 potential).
Potential difference => Current will flow until the charge on both plates is the same.