How to edit 16-bit and HDR images for free
Paint by Zaweba has 16-bit and 32-bit documents, opens EXR and Radiance .hdr files, and runs its built-in filters at 16 bits per channel.

Editing 16-bit and HDR images costs nothing in Paint by Zaweba. File > New has a bit depth setting and Image > Mode switches a document between 8, 16 and 32 bits per channel. OpenEXR and Radiance .hdr files open as 32-bit documents, RAW files open as 16-bit through the RAW Developer, and the built-in filters all run at 16 bits.
Bit depth is about how many steps of tone each colour channel can hold. Ordinary JPEGs have 256 per channel, which is fine to look at but runs out quickly when you stretch the tones in an edit. That's when skies start to show stripes, which is called banding.
16-bit for everyday photo editing
A 16-bit document has 65,536 levels per channel. You won't see the difference on screen at first, but after strong Curves, a big exposure change or heavy colour grading, the 16-bit version stays smooth where the 8-bit one breaks up. For photographers working from RAW, 16-bit is the sensible default.
Switch an existing document with Image > Mode; it takes one undo step, so you can go back if needed. Converting an 8-bit JPEG to 16-bit doesn't invent detail that wasn't there, but it does give your edits more room so they don't add banding of their own.
32-bit and HDR
A 32-bit document is linear light with values above 1 allowed. That means a bright window or the sun can be many times brighter than white and still keep its detail, instead of being clipped. EXR files from 3D renders and Radiance .hdr files open this way, and multi-part EXR files open as layers.
You can make your own HDR picture too. File > Automate > Merge to HDR combines bracketed exposures, with ghost removal for things that moved between shots, and saves .hdr or .exr.
At 32 bits there are 32 filters on offer and some tools are greyed out, because not every operation makes sense on values brighter than white. A common workflow is to do your heavy tonal work at 32 bits, then switch to 16 bits for the finishing touches.
Exporting without losing the work
When you export to an 8-bit format such as JPEG, Paint dithers the result, which hides banding in smooth areas. PNG can be written at 16 bits and TIFF at 8 or 16, so choose those when the file is going on to more editing or to print. Keep the .ptx as your master, since it keeps every layer at full depth.
Mistakes to avoid with deep colour
Don't flatten and re-save as JPEG between editing sessions. Each save throws away the extra depth and adds compression. Keep working in the .ptx until you're finished.
Don't expect a 32-bit file to look right straight away. Linear light looks flat and dark on a normal screen until you adjust it, which is expected, not a fault. Use Curves or exposure adjustments to bring it into a pleasing range.
Colour management matters more at high bit depth too. Edit > Colour Settings lets you choose a wide working space such as Rec. 2020 or Display P3, and exports embed the profile so the colours carry across.
Paint by Zaweba is free and works offline, so you can handle big EXR and RAW files without uploading them or paying for a subscription.