1. A thermostat, comprising: a plurality of HVAC (heating, ventilation, and air conditioning) wire connectors configured to receive a plurality of HVAC control wires corresponding to an HVAC system; a thermostat processing and control circuit configured to at least partially control the operation of the HVAC system; and a powering circuit coupled to the HVAC wire connectors and configured to provide an electrical load power to the thermostat processing and control circuit, the powering circuit comprising: a power extraction circuit configured to extract electrical power from one or more of the plurality of received HVAC control wires up to a first level of electrical power; a rechargeable battery; and a power control circuit coupled to the power extraction circuit, the rechargeable battery, and the thermostat processing and control circuit, wherein the power control circuit is configured such that: (i) during a first time period in which the electrical load power required by the thermostat processing and control circuit is less than said first level of electrical power, the power control circuit supplies the required electrical load power to the thermostat processing and control circuit and charges the rechargeable battery, if needed, using power from the power extraction circuit; and (ii) during a second time period in which the electrical load power required by the thermostat processing and control circuit is greater than said first level of electrical power, the power control circuit discharges the rechargeable battery and supplies the required electrical load power to the thermostat processing and control circuit using a combination of (a) power from the power extraction circuit, and (b) power from the discharging rechargeable battery.