NOVELTY - Power system (16) comprises a rechargeable energy storage (30) that is operable at -40 degrees C to 210 degrees C coupled to a circuit (58) for at least one of supplying power from the energy storage and charging the energy storage, where the energy storage is configured to store between one hundredth (0.01) of a joule and 100 megajoules of energy and to provide peak power of one tenth (0.10) of a watt and 100 megawatts, for at least two charge-discharge cycles.    USE - The power system is useful: for supplying power in high temperature environment; to supply power to a logging instrument (coring tool, shut-in tool, nuclear magnetic resonance imaging tool, electromagnetic telemetry tool, mud-pulser telemetry tool, resistivity measuring tool, gamma sensing tool, pressure sensor tool, acoustic sensor tool, seismic tool, nuclear tool, pulsed neutron tool, formation sampling tool or an induction tool) (claimed); and for instrumentation and tooling used in subsurface environments generally in the exploration for hydrocarbons.    ADVANTAGE - The power system: provides increased level of safety and reliable power in environmentally harsh conditions, limited space available for containing energy storage and substantially remove location where it is difficult to communicate with external power supplies; containing energy storage, which does not contain substantial quantities of lead or lithium; includes extension of useable battery life by limiting temporary voltage dips normally seen during high current draw; provides power supply, which adds flexibility with inherent power conversion so batteries may be more easily interfaced with tools and instruments downhole; provides boost power while smoothing and limiting battery current draw using high-temperature rechargeable ultracapacitors; and eliminate effects of voltage variations on direct current bus that may lead to issues comprising cross-talk or unwanted coupling across signal wires, by reducing large, fast voltage deviations from electrical circuits with in the power system. The power system utilizing ultracapacitor, which: exhibits less internal resistance for similar volumes and weights when compared to chemical battery energy storage technologies, thus enables higher power delivery capability and higher power recovery capability; reduces the need for other energy storage technology and reduces the needed handling, safety and disposal logistics required to support downhole applications; and can store 50-1000 times more energy than capacitors of comparable size and volume, thus enables high power for a longer duration in downhole tools.    DETAILED DESCRIPTION - INDEPENDENT CLAIMS are also included for:    (1) providing power to a logging instrument (10) downhole, comprising selecting a logging instrument that comprises the power system, and with the logging instrument downhole, providing power from the power system to the logging instrument;    (2) fabricating the power system for a logging instrument, comprising selecting the rechargeable energy storage and configuring the energy storage for incorporation into the logging instrument;    (3) using a power supply, comprising selecting a power supply that comprises at least one ultracapacitor and operating the power supply within the temperature of -40 degrees C to 210 degrees C while maintaining a voltage of 0.1-4 volts on the ultracapacitor for at least one hour, where at the end of the hour, the ultracapacitor exhibits a leakage current less than 1000 mAmp per liter of volume over the range of operating temperature; and    (4) using the power system, comprising coupling the rechargeable energy storage configured for high temperature operation with electronics configured for high temperature operation and operating the power system by withdrawing pulses of power from an output of the power system, where each pulses comprises a peak value of at least 0.01 W and total power-time product (energy) of at least 0.01 J.    DESCRIPTION OF DRAWING(S) - The figure shows power system that includes a high temperature rechargeable energy storage.    Logging instrument (10)    Power system (16)    Energy storage (30)    External energy supply (51)    Control circuit (58)