How to stop long file names being cut off on a disc
Burn by Zaweba keeps long and Unicode file names on data discs (up to 127 characters on UDF) and warns you before it would shorten any name.

Burn the disc with Burn by Zaweba and use a layout that includes UDF. Burn keeps long and Unicode names, with UDF holding names up to 127 characters, and it warns you before anything is shortened, so you can fix a name or change the layout before the disc is written.
Cut-off names are an old disc problem that still catches people out. You burn a folder of carefully named files and, on another computer, half of them end in a tilde and a number.
Why names get shortened
Discs use their own file systems, and each has a limit on name length. The ISO 9660 + Joliet layout that Windows writes keeps names up to 108 characters. Anything longer is given a shortened form ending in ~1. UDF allows names up to 127 characters. The full path matters too, since folders nested deep with long names add up.
Some burning programs shorten names quietly. You only find out when you look at the finished disc, and by then a write-once disc can't be changed.
Burning with names kept whole
Open a new data disc from Home, Disc > Data disc or Ctrl+N, and drag your files in from Explorer. In the options, choose the file system. ISO 9660 + Joliet + UDF gives the widest reading with UDF's longer names for computers that understand it, and UDF alone is fine for discs only modern PCs will read.
If any name is too long for the layout you've chosen, Burn warns you before burning. That gives you two easy fixes: rename the file to something shorter, or switch to a layout with UDF. Unicode names, such as those with accented letters or other alphabets, are kept as they are.
Burn also switches to UDF by itself when a file is bigger than 2 GiB, which ISO 9660 + Joliet can't hold, and shows a notice when it does.
Arranging files so names stay short
Sometimes the cleanest fix is to change the structure, not the names. Deep folders with long names make long paths. Turning on Advanced mode, from the switch in the status bar, View > Advanced mode or Settings, gives new data discs a project window with your PC's files on the left and the disc on the right. There you can make your own folders on the disc, rename items and drag things into a flatter layout without touching the originals on your PC.
Checking the result
With per-file verify on, or the setting Check every disc after burning, Burn reads the disc back after burning and compares it with the original, so you know the files arrived intact. Opening the disc in Explorer afterwards is a quick way to see the names for yourself.
If the disc is for older equipment such as a car stereo or an old player that only understands the simplest layouts, keep names short from the start. For backups made with the Backup Assistant, you don't need to worry: Burn plans the discs and keeps an index of every file, and a restore puts them back with their names.