Does a laptop recharger waste electricity when the laptop is not connected

The answer for this is: YES, it will consume power, but such a little power that you may consider it irrelevant in terms of overall power consumption.

As you can see in the picture below (a schematic of an AC notebook adapter), the "entrance" of the circuit (“L and N 90 - 265 VAC”, in the lower left corner) has an AC transformer:

This component is responsible for converting the entrance AC voltage into another one in AC (if there is a load on the "output" side of this component). As you can see in the picture below:

Wikipedia has a good explanation about the operation of a transformer:

A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field through the secondary winding. This varying magnetic field induces a varying electromotive force (EMF), or "voltage", in the secondary winding. This effect is called inductive coupling.

If a load is connected to the secondary, current will flow in the secondary winding, and electrical energy will be transferred from the primary circuit through the transformer to the load. In an ideal transformer, the induced voltage in the secondary winding (V_s) is in proportion to the primary voltage (V_p) and is given by the ratio of the number of turns in the secondary (N_s) to the number of turns in the primary (N_p) as follows:

I marked in bold the main part of explanation to explain what will happen if there is no load on the transformer (in this case, it means that there is no device "consuming" power from the transformer). If there is no load, the secondary part of the transformer will not "act" on the primary part of it.  It means that you will only have the input loaded with the entrance impedance of the transformer, or, in this specific case, the AC power from your electric outlet to an inductor of 200 µH in the transformer. A theoretical "ideal" inductor will not dissipate any power, but a real inductor will also have a resistive component that dissipates a small amount of power, usually due to the magnetic material in the core of the transformer producing "core losses". This will explain why you have a consumption of the AC adapter even if there is no "load" or on this case, if the notebook is turned off.

PS: This situation applies not only for your notebook AC adapter but to all adapters that have a transformer on their entrance.

Hope being helpful with my explanations.

No testing needed. Nothing wrong with the outlet can do this in a double insulated device. Replace your charger, ASAP.

This is a safety issue, because it only takes 30 milliamps of 60 Hz current across your chest to stop your heart...

In the United States, the Occupational Safety and Health Administration (OSHA) has published an online brochure that makes the point that grounding won't save you. A grounded (earthed) outlet prong only works if the insulation fault in the equipment is fully connected to a proper ground (cold water pipe or ground stake deep enough in the earth). If you're a better conduction path than the ground conductor, you get the shock instead of the ground wire.

Better than grounding is double insulation -- as long as the device isn't tampered with or damaged, a double insulated device can't shock unless it gets wet or is exposed to conductive dust (metal shavings, etc.). These are recognized by a two-conductor power connection, and often an all-plastic housing.

The only dependable protection (according to OSHA) from insulation fault shocks is a Ground Fault Circuit Interrupter (they go by different names in other countries). These detect tiny differences in the current on the hot and neutral wires (ignoring the ground wire) and shut off the circuit if the two aren't equal. Before you can perceive a shock, the GFCI will shut off the power to the device and trip its warning (usually a visible button popped up, similar to a circuit breaker showing tripped). That's your cue to replace the device, or get it repaired by a professional.