Yes, but this is exactly the sort of thing that should be a user-level configuration. A personal scratch directory has nothing to do with the repository itself and doesn’t belong in a repo’s .gitignore.
If it's only needed for one particular checkout, .git/info/exclude is the other obvious option.
You don't need to put the scratch directory in the committed gitignore file. gitignore files are recursive. So a common approach is to use a gitignore with * in the scratch directory.