With the advent of helium-filled drives, manufacturers are coming out with 10 TB, 12 TB, and even 14 TB hard drives that spin at 7200 rpm, enabled by the lower density of helium which reduces turbulence especially with lots of platters inside the drive. (Previously, larger drives would be limited to 5400 rpm, and traditional non-helium hard drives are currently limited to 8 TB.)
As hard drive capacities increase, their speed increases as well, and this means that electromechanical disks can saturate the SATA 3Gb/s interface. In particular, the Seagate BarraCuda Pro 12 TB drive can exceed 270 MB/s. This is very close to the theoretical 300 MB/s allowed by SATA 3Gb/s, and in fact, is close enough that the drive may not be able to reach this speed in practice without a SATA 6Gb/s link.
However, there is currently no hard drive capable of reaching the 550-560 MB/s or so needed to saturate the SATA 6Gb/s interface—the aforementioned BarraCuda Pro is only halfway there. As hard disk technology advances, however, there may very well be drives that can hit these kinds of speeds. HAMR and MAMR are expected to enable hard drives to exceed 20 TB or even 40 TB in capacity. These drives will most likely be released for datacenter use first, where the predominant interface is SAS 12Gb/s, twice as fast as SATA 6Gb/s. However, there is precedent for enterprise storage technology reaching consumers later on (this has happened with both NVMe SSDs and helium hard drives), so chances are good we will eventually see drives that can saturate SATA 6Gb/s. By the time these kinds of drives come out, however, it's very likely they'll get a faster interface to support these speeds.
In short, there are electromechanical hard drives that can require SATA 6Gb/s to attain full performance, but there aren't any that can saturate the interface.
On the other hand, the vast majority of SSDs are capable of saturating the SATA 6Gb/s interface, and the fastest consumer PCIe SSDs can operate at up to six times the sequential I/O speed of SATA SSDs (and this is from my personal desktop, Astaroth):