Electronic – What’s the point of a current source

An open circuit, by definition, can have no current. A current source, by definition, has a non-zero current. The two are in series so the currents must be equal, but can't be. You have created an impossible circuit.

The dual of your impossible circuit would be an ideal voltage source into a short:

^{simulate this circuit – Schematic created using CircuitLab}

A short, by definition, can have no voltage across it. A voltage source, by definition, does. The two are in parallel so the voltages must be equal, but can't be. This circuit is equally impossible.

You could say that the current in this circuit is infinite, and the the voltage in your circuit is infinite. What's the power in those circuits?

$$ P = I E = 1A \cdot \infty V = \infty W$$

$$ P = IE = \infty A \cdot 1V = \infty W $$

The power is infinite. What does that even mean? I have no idea: you will have to ask a mathematician.

If you were to consider that any two separated conductors are a capacitor, then maybe the circuit you had in mind was this:

^{simulate this circuit}

Then yes, the voltage across C1 will increase, linearly, forever.

There is no return path from ground back to the other side of the source, therefore no current will flow to ground. All of the current will flow through the resistor.