BRAIN - every inspector needs one, not just to read and memorize a lot of stuff, but to also try to understand what he's looking at and how it affects the other components and systems, plus simultaneously explain to and converse with the client. 

BODY - must be reasonably fit to access roofs, crawl spaces, operate other tools, etc.

COMPUTER - Very handy for checking manufacturer's websites for build dates, recalls, accessing inspector's forums, continuing education classes,  email communications, and creating  and uploading our  full color digital reports.

VEHICLE - reliable, able to transport tools, ladders,etc.

BINOCULARS - for some aspects of roof inspection; mine are 12-20 x 50's; most roofing materials are at least slightly damaged with foot traffic, some roofs are too steep to mount without special equipment, some are ice or snow covered, some have to be inspected on very windy or rainy days, and some buildings are too close to each other to allow a safe ladder position.

DIGITAL CAMERA -good for note taking, documentation, and invaluable for clearly communicating your findings to various parties. 

GOOD FLASHLIGHT - because dark places need inspecting too.

DROP LIGHT - for flue and chimney inspections from above.

ELECTRICAL TESTER - quickly diagnoses many wiring problems behind the cover plates of outlets such as open ground or hot/neutral reverse. These outlets work ok but can damage sensitive electronics or be unsafe in certain circumstances.

ELECTRICAL PROBE - to safely verify power/no power of different objects.

BORE SCOPE - narrow cable with a camera and light on one end and a monitor on the other; for inspecting  behind walls, etc.

INFRA RED CAMERA - picks up cool or warm areas of a structure; used for finding water leaks, [=mold ?], insulation gaps, etc.

INFRA RED THERMOMETER - simpler version of the above camera for finding non-contact temperatures; these readings can predict equipment failures, etc.

 BATTERIES - and charger for other tools.

METERS - for measuring components for electrical and  moisture properties.

LADDERS - because you can't do it all from the ground.

NERVES OF STEEL - some creepy places need inspecting.

COOKING THERMOMETER - to take the temperatures inside the ductwork and to check the hot water.

SCREWDRIVER, HANDTOOLS - important to inspect some things which have covers even though inspectors are not required to remove covers; sometimes curiosity gets the best of you or you just need it for thoroughness. Also to do the occasional minor repair as a good neighbor.

MICROSCOPE - portable  type; can assist in identifying some insect residues, effluorescence crystals, etc.

STETHOSCOPE - mechanics type; useful to quickly locate equipment failures, etc.

WATER PRESSURE GAUGE -  to measure for adequate delivery pressures, or document poor ones.

GAS SNIFFER - to find leaks in gas lines, CO , backdrafting, etc.

LEVEL/PLUMB BOB - to verify sanitary plumbing slopes, document basement wall deflection, etc.

TAPE MEASURE - for handrail heights, stair tread and riser variations, clearances,sizes, etc.

SMOKE PEN - for rough air flow evaluations at registers, flue vents, radon diagnostics, etc.

MANOMETER - for pressure measurements related to radon diagnostics and testing.

POWER TOOLS - including diamond core drill bits, for radon system installs.

CRM - continuous radon monitor, for accurate and reliable radon testing.

PPE - personal protective equipment, like respirators, face shields, ear cups, safety glasses, gloves,etc. Even a tyvek suit and duct tape for the rare really nasty inspection.