The most extreme example is series resonance, although your question is equivalent to asking how circulating current can be much larger than net current draw for parallel resonance. For series resonance, the voltage at the LC junction can be vastly larger than that of the driving source. It is also 180 degrees out of phase with it. At resonance, the L and C impedances are equal and opposite (+j and -j) and therefore cancel, leaving only parasitic resistances and driver impedance to limit current. Nevertheless, the current through each element creates a voltage proportional to the impedance of each individually, which as noted can be much larger than that driving the circuit. The fact that the voltage at the junction is out of phase with the driving source is how it is possible to draw current higher than either element would on its own.
Interrupt a dc current running through a large inductor and sparks will fly. v = L di/dt as I recall becomes quite large as the current drops to zero.
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