Count only filled, nonblank cells

I'm having a problem with coming up with a good way to count nonblank cells. Excel doesn't seem to have that function. The ones I can find are:

Count, which Counts Numbers
Counta, which Counts Numbers, Text, and Formulas
Countblank, which Counts "empty" cells

The problem is, say I have a range of 5 cells containing a number in one cell, Text in another, a blank in the third, and formulas in the last two that trigger based on the value in A3, either with a text or number response, or "" to mean "nothing". None of the functions above can tell me how many cells actually have values in them at any given time. I need a COUNTA that DOESN'T count formulas.

I've attached my sample book. The only solution I found that appears to work is this:

=(ROWS(A1:A5) * COLUMNS(A1:A5))-COUNTBLANK(A1:A5)

I'm wide open if there's another way to do it. Seems like a simple thing, but not for me.