The function to show current file's full path in mini buffer

It's the built-in function buffer-file-name that gives you the full path of your file.

The best thing to do is to have your emacs window to always show your system-name and the full path of the buffer you're currently editing :

(setq frame-title-format
      (list (format "%s %%S: %%j " (system-name))
        '(buffer-file-name "%f" (dired-directory dired-directory "%b"))))

You can also do something like this :

(defun show-file-name ()
  "Show the full path file name in the minibuffer."
  (interactive)
  (message (buffer-file-name)))

(global-set-key [C-f1] 'show-file-name) ; Or any other key you want

To borrow from Jérôme Radix's answer, if you just want to quickly see the file path of the current buffer, you can do M-: buffer-file-name.

Alternately, type (buffer-file-name) in the buffer somewhere and run C-x C-e on the closing parenthesis (this will work in any mode, not just lisp-mode).

My trick is to do a C-x C-f like to open a file, it wil prefill the minibuff with the current file path, C-g to quit. Faster than M-: buffer-file-name but far far uglier than any other methods.