"increasingly great" is correct and can be used in written English.
It is usually used to describe something that is growing or increasing in size, extent, or intensity. For example, "Sales of this product are becoming increasingly great as more people become aware of its advantages."
It's another classy effort from an increasingly great band.
"The level of JGBs that the banks have is becoming of increasingly great concern.
Yet since the outrage over the expenses scandal, it is likely that normality will be an increasingly great asset.
With insects in natural habitats, competing not only with members of their own species but with numerous other species as well, the obstacles to survival become increasingly great.
It's a nod to the ridiculous luxury of space that you expect to see only in Berlin, but that even the Murkudises of the world must go to increasingly great lengths to find.
Now, songs such as Restless Year just sound increasingly great, rather than ersatz, and I listen to Furman cook an egg on Haunted Head and well up ("I get a slice of bread/I cut a hole in it/I crack a little an egg into it/Frying pan/And I try to get my mind turned off... ..).
They are concerned about the future for their children, grandchildren and increasingly great-grandchildren.
The party broadened its base to attract increasingly greater segments of the middle class.
But instead of being deterred, the fighters who show up each day face an increasingly greater risk.
Despite this overwhelming shareholder support, corporations are going to increasingly greater lengths to ensure they pass the vote and avoid the negative consequences of a failure.
The 'new normal' expects students and their families to continue to make increasingly greater financial sacrifices in order to complete a postsecondary education.
