Manual:$wgFileExtensions

Details
This is the list of preferred extensions for uploading files. Uploading files with extensions not in this list will trigger a warning when $wgStrictFileExtensions is false. However $wgStrictFileExtensions defaults to true, blocking any file extensions not in this array. This option does not have any effect if $wgCheckFileExtensions is false. Some filetypes are on a blacklist: $wgFileBlacklist.

Some files, for example documents Open Office documents use mime types that don't match with their extension. These files uploads result in an error message like ''The file is corrupt or has an incorrect extension. Please check the file and upload again.''. In such cases you can adjust the mapping of file extensions to mime types, see MIME type validation.

Comment
To allow uploading of files that don't have extensions, edit $IP/includes/SpecialUpload.php and at line 351 (in version 1.11.1), change

to In other words, comment out the action that is taken if there's no file extension.