Archimedes said, “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.”
This is the nature of the thought exercise you’re asking. And the answer is, yes, given a suitable conversion system (our fulcrum and lever, so to speak), you could ‘run everything in the world’, at least briefly, with the familiar 9V (PP3) battery.
Without getting hung up on the practicalities, energy is energy, power is power, and a 9V battery certainly has some.
Now, consider that a 9V battery is a pretty small (and non-renewable) packet of chemical energy compared to, say, an off-grid battery bank, a generating plant, or for that matter, the entire world’s power grid.
How small? A typical alkaline 9V battery is good for about 400mAh, or about 3.6 Wh (0.4Ah * 9V) of electrochemical energy from that little guy.
Compare that to the total global energy production, which in 2021 was about 28,000 TWh (link). This works out to about 3.2 TW of power being used worldwide at any given moment (28,000 TWh/365d/24h.)
In round figures then, it would take about one trillion 3.6 Wh 9V batteries per hour to supply the global energy demand. Or, one 9V battery could supply one trillionth of that demand. Or it could supply it for one trillionth of an hour, that is, 3.6ns (as noted by @Neinstein).
You could do similar analyses for, say, a whole household or a even single appliance, and come away similarly disappointed with the 9V battery’s ability to power anything but the smallest of devices.
And that’s okay. The 9V battery isn’t designed for bigger stuff. Internally, 9V batteries are actually a stack of six tiny 1.5V cells connected in series. The internal resistance of that stack is pretty large, severely limiting the current that the battery can deliver.
Rather than powering your house, or the world, the 9V battery’s mission is to power small electronics like ‘transistor’ radios: that is, devices with a small current drain but in need of a higher voltage to operate. And for that it’s pretty darn good, if pricey.
The 9V battery: great for your smoke detector, not so much your toaster oven making the smoke.
You might do slightly better if you Mr. Fusion’d the battery, converting its entire 45g mass to energy. Then to ‘power the world’ you’d need to torch just three per hour: using e=mc^2, each battery yields 4.05 x10^15 joules (4.05 petta J, PJ) or 1.125 TWh.
(1 J = 1 watt-second; 1 watt-hour is 3600 watt-second or 3600 J.)
Fun fact: the largest US nuclear test, Castle Bravo (15 Mt), released about 62 PJ of energy, or roughly sixteen 9V batteries’ worth of mass into energy. Castle Bravo's energy output would power today’s world for about 3 hours. I suppose you'd have to upgrade to Tsar Bomba (50 Mt, 209 PJ) to get longer play time.
Realistically though, both science and economics limit the 9V battery use cases. A 9V battery is heinously expensive per unit of energy delivered, very resource-intensive to make, non-rechargeable in its common form, and limited in both energy content and power output.
Sadly, as things stand today most alkalines end up in landfills. Only about 4% are recycled at present, although there are efforts underway to improve this (link).
Li-ion and related chemistries come closer to meeting the need for scalable energy storage, which is why they're the go-to choice right now. But even these have problems when viewed as a life-cycle. Li-ion recycling is only now being looked at seriously (link).
So what else can a 9V battery do? How about rewiring your brain? Yes, it’s a thing. (Disclaimer: you can hurt yourself doing this. Not recommended as a DIY.)