How does energy dissipates in a purely reactive circuit?
If there is GENUINELY ZERO resistive energy then energy will not dissipate, an ideal energy meter will not increment and the ideal reactive components will maintain the situation indefinitely.  

Then, you wake up.
In the "real world" there are NO genuinely reactive components.
Inductors can resistance (and some capacitance).
Capacitors have ESR (resistance) and some inductance.
Wires have the same.
The energy meter shunt is often resistive, and even if it uses a Hall sensor it will not be absolutely lossless.  

In the very very "best" case, it may take a wee while for things to die down. But never anything like indefinitely. You can work out the order of time for yourself by looking at eg published capacitor ESR values and inductor resistance, and wiring resistance. These set a lower level on losses before you start to look at the other (usually) second order components.