How can I get this HDR or painted blur effect from a photo in Photoshop, Gimp, Inkscape or some other software?
To create a similar effect as can be seen from HDR photos we can increase overall image details by a fake contrast enhancement by a dynamic range increase.
There is quite a cool effect "Freaky Details" from the G'MIC plugin to Gimp which tremendously helps us to do so.
As Scott mentions indirectly, this is what is usually called an HDR photo. The idea is you take a series of photos by bracketing: you take a single good photo and then 2 or more by altering the exposure (time or aperture) in both "directions" (lighter and darker). HDR requires a minimum of three I think.
The overexposed ones give you more detail in the shadows, and the underexposed ones give you more detail in the lightest areas.
Then you use software which can handle HDR (Photoshop and GIMP both do AFAIK at least, with plugins). This creates an HDR image which you manipulate and then save as a standard 8 bit per channel RGB.
The halo effect you see in this one is probably due to fog or moisture in the air coupled with over-exposure.
Note that some cameras have HDR automation (autobracket and postprocess) built in and some smartphones cave apps for it.