Sure...anything's possible if you have the finances. A monolith slab can be had. Stone yards have them for counter tops. Some of the oversized slabs are in the 8ft range. The question isn't can it be done, the question is should it be done?

A single stone is SLICK. As in "arse over tea kettle" if anyone is wearing heels (90% of men don't understand this one...). Part of a grouted floor is the grip.

Terrazo would be a great option. It will be expensive. This old fashioned floor has been brought back into vogue by using high-value acrylic/epoxy setting. It is gorgeous and wears very well. It is hard to find in a residential setting because of its price tag and the lack of professionals who are trained in it. The people who do it are normally in the commercial building side of things. They rarely do residential because they don't like dealing with homeowners (sad but true).

When looking at monolithic size tiles or stone, be aware that many of these fail because of the lack of talent and skill on this side of the Atlantic. The Italians and Spanish have plenty of workers who can do this type of install....but not many N. American companies will send their guys for training weekends to Barcelona or Milan.

Grout is a requirement for tile tile installations. It provides allowance for off sizing and building movement. Super large format tile or a stone slab is always a possibility. but the logistics and budget have to work for $100 a square plus+. A entry like you describe would almost have to have that laid first and then the house built around it. The likelihood of any of that cracking are directly proportionate to choosing the guy who promises he can do it for $50 a square vs. the trained and experienced guy who knows the difficulty level and charges $100+.