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Batteries with a useful and durable energy storage capacity, so that we don't have to keep recharging our watches/phones/laptops/cars every ten minutes.

I’m not sure about that, I don’t know anyone who has to charge their phones ‘every 10 minutes’ (Unless they require a battery replacement), especially given how large batteries are in phones today, I think the battery capacity is more than sufficient for the demands for most in terms of media/communication.

However, lithium ion battery technology is dated, but it has improved a lot over the years with the help of the efficiency of the displays and processors. Next Gen battery tech is still likely years out, which I suspect we will see micro-LED before any battery upgrades.
 
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I’m not sure about that, I don’t know anyone who has to charge their phones ‘every 10 minutes’ (Unless they require a battery replacement)

Sometimes, people exaggerate to make their point ;-)

And I would love more battery life! Stop making our iPhones thinner and thinner. It's ok, their thin enough, now start putting in more and better batteries!
 
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Sometimes, people exaggerate to make their point ;-)

And I would love more battery life! Stop making our iPhones thinner and thinner. It's ok, their thin enough, now start putting in more and better batteries!

I don't see better battery life in phones as the "next quantum leap that will affect our everyday lives" (my question) but rather better battery life having significant impacts in electric vehicles and as large-scale storage/backups for green energy sources.
 
The Japanese did:
https://www.nytimes.com/2009/07/20/technology/20cell.html
https://en.wikipedia.org/wiki/Mobile_phone_industry_in_Japan

Not that there weren't other companies playing with these ideas (IBM and Ericsson stand out):
https://www.webdesignerdepot.com/2009/05/the-evolution-of-cell-phone-design-between-1983-2009/

But Japan was leading adoption.

We assumed that the Palm Pilot and phone would merge at some point, but I'd visit Japan and wonder why the heck anyone would want to take a picture with a phone. But they were using their phones for everything-- email, SMS, pictures, games, subway payments, mini-web pages, TV, radio... I always assumed it was because they wanted to make use of the time spent on subways.

I suppose it depends on what you mean by "iPhone". Touch screens and cameras were around or nearly around 20 years ago, just crappier versions of them.

The tech has improved, and some surprises popped up, but for the most part we keep improving on old ideas until they become usable. I expect we'll see the same trend over the next 20 years. Ideas that seem underdeveloped today, will be streamlined and so deeply integrated into our way of life that it'll feel revolutionary even if the ideas themselves date back to the 80's:

https://en.wikipedia.org/wiki/Virtual_reality
or before:
https://en.wikipedia.org/wiki/Artificial_intelligence#History
Lo and behold we are now clamoring for paying for subway fare with our phones in the US, 20 years later, LOL! To hold you over, at least you can use Apple Pay to buy a subway sandwich until OMNY rolls out, LOL!

You can already use Apple Pay to get on trains in Chicago and Portland, and many agencies have apps for the same purpose (not inside of Apple Pay, but uses Apple Pay to buy the Pass, like San Diego, and SF)
 
If you watch the original Star Trek series and then Next Generation, you will see lots of gadgets that were dreamed up long ago being used now
So yes, 20 years ago what we see today was already there in the minds of some folks

My biggest tech influencer was Star Trek reruns followed by a lot of old 80's/90's cartoons and sci-fi novels. I've been waiting on a lot of tech to become reality for a long time.

I remember the Star Trek reruns and wanting a communicator so badly. I was excited for the Star Tac and went nuts over the Motorola Razr - dreams come true!!! Plus the pads that the engineers or the Captain would carry around. Yep, had to get an iPad when it was introduced! Inspector Gadget with the watch - got an Apple Watch.

Waiting on interactive holographic renders (Minority Report/Iron Man/etc) and AI assistants that are really useful that I can have true conversations with as I'm giving commands (without going Terminator, of course). Alexa/Siri - just not doing it for me.
 
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Lithium is already the battery leap we needed. We just want yet another one.

The next big leap in battery design is something like solid-state batteries that don't use dangerous ingredients like lithium. There'll be incremental improvements to lithium-based designs but they won't be a technological leap.
 
Like a few people here have said, shows like Star Trek predicted smartphones and tablets. And there is even what very much looks like an iPad in 2001:a space odyssey (1968!).

I remember in 1999 my focus was the web being the revolution in itself and for me, things like broadband being on the horizon were more game-changing.

Hardware-wise, the early to mid years of the last decade in mobile seemed very stagnant, with only the iPod and iTunes music store feeling truly revolutionary. PCs didn’t change that much. Linux being truly user friendly on the desktop never really materialised. Microsoft were lost in the mess of vista. Apple were preoccupied with building out OS X. the web had exciting things happening in the form of Ajax and Web 2.0 but Flash was starting to strangle all of that.

So when the iPhone came along it truly was revolutionary and kick started a whole host of other tech (the uptake of 3G, mobile apps, the collapse of flash etc).

So I like to think I knew that something like the iPhone was coming. But I know at the time I never guessed how industry(s) changing that it would be.
 
Now a totally different view. You asked what we thought in 1999. Now a very good read is Harry Harrison's "1999" (first published 1966) where he predicted what the world would look like in 1999.

US population over 500 million. Roads being pulled up to make more space for farming land to feed a starving populations. Cops in New York can't get any new ammunition anymore because none is manufactured, but taken from storage. With "Manufactured 1960" written on it.

20 years later, and we are still not there yet. Will probably take another 50 years. You can still have children (but not too many in the USA, because according to the US president, there is no space).
 
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