vim: number of total buffers

Answers so far are too hacky. Here is vim's built-in way:

len(getbufinfo({'buflisted':1}))

As always, see vim's help (:h getbufinfo()) for the official explanation.

Same idea than Heptite's solution, but as a one liner. Many other things may be done this way: get the name of the buffer (thanks to map), wipeout buffers that match a pattern, https://stackoverflow.com/questions/2974192/how-can-i-pare-down-vims-buffer-list-to-only-include-active-buffers/2974600#2974600n etc.

echo len(filter(range(1, bufnr('$')), 'buflisted(v:val)'))

To my knowledge there is no built-in method in Vim to do this, but you could create a function:

function! NrBufs()
    let i = bufnr('$')
    let j = 0
    while i >= 1
        if buflisted(i)
            let j+=1
        endif
        let i-=1
    endwhile
    return j
endfunction

Put the above in a text file with its name ending in .vim, :source it, then you can do something like:

:let buffer_count = NrBufs()
:echo buffer_count

June 21 note: If you have a recent version of Vim as of 2017, Gid's answer below is the optimal solution.