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how many times nestwhile repeats calculations?


>From Help Browser we read

NestWhile[f, expr, test] starts with expr, then repeatedly applies f
until applying test to the result no longer yields True.

I would like to know how many times f is applied till test stops
everything. I read already a post in which the following example was

In[1]:= mysqrt[a_] := NestWhile[{#[[1]]+1, .5(#[[2]]+a/#[[2]])}&,
                                {0, Max[1.,N@a]}, #[[2]]^2 > a&]
In[2]:= mysqrt[2]
Out[2]= {5, 1.41421}

Do you know any other method? I don't want to influence to NestWhile
(by inserting a "counting" element #[[1]]+1 )



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