Calling completion-show-menu when completing, bound to M-Down, will bring up the completion menu, from which you can select a completion to insert. If completion-use-hotkeys is enabled, the menu will also display the hotkeys next to the completions they select, although you will have to exit the menu before you can use them (see Completion Hotkeys).
The completion menu also contains a Browser
entry, which replaces
the completion menu with the completion browser. This hierarchically
lists all possible completions, irrespective of the setting of
completion-max-candidates (see Miscellaneous Options). If it's
taking too long to construct the browser, you can hit C-g to
cancel it. You can also display the browser directly using
completion-show-browser-menu (not bound to any key by
default).
The completion menu can be displayed automatically when completing, optionally after a time-delay, though because it steals the keyboard focus this is less useful than auto-displaying the tooltip or pop-up frame. See Auto-Show a List of Completions.
The following variables affect the behaviour of the completion menu and browser:
balance
causes the number of entries in all menus,
submenus, subsubmenus etc. to be made as equal as possible. The symbol
max
maximizes the number of entries in higher level menus (and
hence minimizes the number of entries in lower level submenus), whereas
minimize
does the opposite. Note that none of these options
affect the number of levels of submenu required in a given browser
instance.