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.

The direct implementation of what you want is:

(defun copy-full-path-to-kill-ring ()
  "copy buffer's full path to kill ring"
  (interactive)
  (when buffer-file-name
    (kill-new (file-truename buffer-file-name))))

That said, I find it incredibly useful to be able to get the full path of what is in the minibuffer, and this is what I use:

(define-key minibuffer-local-completion-map "\C-r" 'resolve-sym-link)
(defun resolve-sym-link ()
  "Try to resolve symbolic links into true paths."
  (interactive)
  (beginning-of-line)
  (let* ((file (buffer-substring (point)
                                 (save-excursion (end-of-line) (point))))
         (file-dir (file-name-directory file))
         (file-true-dir (file-truename file-dir))
         (file-name (file-name-nondirectory file)))
    (delete-region (point) (save-excursion (end-of-line) (point)))
    (insert (concat file-true-dir file-name))))

And then if I want it in the clipboard, I just kill the line (C-a C-k). But we could easily copy the truename to the clipboard in the above command, just change the last line to be:

(insert (kill-new (concat file-true-dir file-name)))))

The new part is the call to 'kill-new which puts the string in the kill ring.