While most of the gadgets may be designed at the time when the house is built, thus power from grid (or from a secondary low-voltage wiring) may be provided, most of the interesting cases is wireless. They feature easy installation, somewhat lower installation cost, and also higher reliability due to no impact of problems of wired power (e.g. there's significantly more HF content in the grid power than it used to be one or two decades ago and who knows  how will this change in the upcoming; houses near modern "renewable" sources such as wind or solar farms are increasingly subject to wild fluctuation of grid voltage; switching power sources are notoriously prone to failure and if such used in low-voltage wiring fails so that it outputs a surge, it may take out the connected devices, etc.). Also, wiring diagrams for a house tend to disappear within decades, and rewiring a failed cable may prove to be exactly as annoying as a failed gadget is.

So, most of the suggested gadgets are wireless, or wireless version of what has a wired version too.

• Clock - clocks are devices with steady but low power demand, so making them completely free from wired power is an attractive option. In this case an appropriate primary power source has to be provided. Clocks typically run out of a single AA cell for roughly a year, given some 4Wh energy content of an alkaline AA cell and slightly below 9000 hours/year, the average consumption is a few hundreds of uW. The current consumtpion in case of LCD display is relatively steady, in hundreds of uA; in case there is a radio receiver for automatic time adjustment there may be periods of higher current consumption, maybe a few mA, for tens of seconds, perhaps once a day. Electromechanically displaying clocks have mA current surges at the moment of  making a tick.
• Thermometer/hygrometer - these usually gadgets with an LCD display, with similar activity thus power profile than the clock, also with a similar consumption. Some of them transmit the measured value wirelessly, so there may be a surge of  power consumption up to a few mA for a couple of seconds, usually once in a few minutes.
• Utility meter (gas, water - electricity is not a problem) check/transmit unit - these usually monitor a dry switch from the locked utility meter, and transmit the consumed amount to some home automation center. Unless thermo/hygro, they don't need a display, but they also have to monitor their input all the time, can't chose when to do that. On the other hand, as the input is a simple switch, their task is relatively simple and the front-end can be designed as of low-consumption asynchronous counter, carefully adding some reasonable means of debouncing. This all may lead to uW static consumption; however data transmission again means an occasional surge of a few mA, with only slightly increase of average consumption.
• Alarm with a dry contact - for example flood alarm, intrusion alarm. They may be almost or completely current-free, until the electromechanical switch is engaged; in which case a one-time or few-times-repeated transmission occurs. A test transmission may be required, say once a day.
• Comfort light - usually a single LED, providing just enough light in the night to move around safely. Requires say ten mW for tens of seconds, maybe a few times during a night. Add a mW continuous consumption if not tripped by a switch, but incorporates a PIR motion sensor.
• Door lock, remote controlled - this is a typically high-powered device, which needs wired power; however, for power outage there must be some plan B, power source for one-time opening (there may be also plan C,  lock having also a mechanical key which is well stored; and/or plan D, a locksmith capable of opening the door while minimizing damage).
• Remote lightswitch - similar power profile than other transmit-upon-switch devices, the difference is that this one is hand-operated.

There are certainly more similar gadgets possible, but for now this will be sufficient as a basic sample.