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.

See also: Manual:Mime type detection

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 taken if there's no file extension.