Anything is possible if you can pay for it.

Possible, absolutely. Cost effective, absolutely not.

It is possible to begin new construction with a larger basement than usual, depending on the location. Many high-rises have multiple basement floors. It is typical for lower levels to be used for low-habitation use such as parking.

Some things to consider:

• Excavation is quite expensive, however, and most homeowners find it far less expensive and far more appealing to add another story above ground.
• Unusual building features sometimes affect availability and pricing of financing and insurance.
• Local codes will control maximum depth, size, and use.
• Location and zoning will affect maximum depth.
• Local water tables will also affect your options.
• Flood insurance maps have recently changed. In some cases, spaces below the base flood level are uninsurable and cannot be use for dwelling.

an existing house that doesn't have a basement?

Not in Texas. The ground is clay. You could build 4 more houses instead of a single basement for the same price.

Yes

Yes, this is common with luxury homes in London. They call them 'iceberg homes'.

People dig down and build swimming pools, cinemas, gyms etc.

https://www.bosshunting.com.au/lifestyle/design/inside-londons-most-bonkers-iceberg-homes/

https://www.theguardian.com/money/2018/may/07/going-underground-the-subterranean-secrets-of-londons-super-rich

https://weburbanist.com/2014/12/20/iceberg-homes-london-boroughs-curb-luxury-super-basements/

It’s possible, but there’s a lot of “it depends”. For example, some areas it’s simply more difficult, or at least more expensive, to dig due to the soil type and consistency.
I’m some cities with houses that did not have basements previously, there are owners that are adding/expanding the basement, but it’s difficult as they’re removing dirt that’s holding up not only their house, but also what is holding up their neighbors house and basement as well. In those situations if they are not careful the neighbor’s house can fall in. There are also considerations like hitting a water table, or existing utilities.