Digital medium format is where digital 35mm was a decade ago. Crop sensors have been introduced to make the digital transition cheaper, but the only 'full frame' medium format sensors are the most recent flagship models in the smallest traditional size.
Medium format sizes
Medium format is pretty much any film or sensor size larger than 35mm (3.6x2.4cm) and smaller than the 4x5 inches of large format.
In the film days, 'medium format' generally meant one of a handful of common sizes - 6x4.5cm, 6x6, 6x7 and 6x9. It turns out, to date, only the smallest of these sizes have ever been approached in a production digital camera. As of time of writing, the largest digital sensor is 5.3x4cm with 100 megapixels, exceeding the previous generation's 50 megapixel 4.4x3.3. This is almost at the 6x4.5cm size (5.6x4.2cm in reality). And it is still a long way from 6x6. Notice in the figure on the right the size of digital medium format (green) compared to traditional film sizes (gold).
35mm's migration to digital
35mm sensors were not very affordable at the dawn of popular digital photography. To drive sales, smaller sensors were made similar in size to APSc film, which ironically faced an early demise due to the growing popularity of digital and the fact it was inferior to a regular 35mm roll in many ways. Camera marketing began using the term 'full frame' for their top-end 35mm-equivalent cameras, and 'crop sensors' for the APSc-sized sensor cameras. Small sizes are nothing new - cameras like the Olympus Pen used a 'half frame' size that allowed twice as many images on a roll of 35mm film. Economy and portability at the expense of image quality. Everything in photography is a compromise.
Medium format's migration to digital
Medium format has no single aspect standard - rather than comparing all sensor sizes to the largest common type, 6x7, like we do with 35mm relative to APSc and micro 4/3, it might only be reasonable to hold the smallest 6x4.5 as the standard for 'full-frame medium format' given this has been recently matched by commercial digital sensors. The larger standard negative sizes still have no competition.
Sensor size is not the full story though. The larger-than-35mm sensors still have an advantage when it comes to colour depth, dynamic range and tonality. A big difference is camera-to-subject distance which affects depth of field.
It is interesting that 'crop vs full frame' comparison is used in the 35mm world but not the medium format one. This might say a lot about the target markets - 35mm buyers care about specs, whereas the professionals using medium format have better things to worry about, like light, creative vision and getting paid.
Full-frame medium format
I'm sure full frame medium format will arrive in my lifetime, perhaps even a 6x6. It might be affordable to the amateur by the time my children have retired. So in the mean-time the only realistic way to shoot full-frame medium format is with a film camera.
Luckily, film is not dead and old medium format cameras can be bought for a fraction of their original cost. From Hasselblads to the Rollei twin lens reflex, Mamiyas and Bronicas. Plenty of film still being made by Ilford, Kodak, Fuji and many others. Film might be too costly and risky for modern professional use, but these are not the concerns of the amateur. If you want to explore truely larger-than-35mm formats, film is definitely the place to start.