Question 1116978
I do not believe ther is such a nonnegative integer, but if you know of a different answer, please enlighten me.

A nonnegative integer is either a positive integer or it is zero.

If {{{r}}} is a positive integer,
and {{{a[n]}}} is an integer term of an infinite geometric sequence with common ratio {{{r}}} ,
{{{a[n+1]=a[n]*r}}} will be an integer,
and so will be every term after that.
As a consequence, there will be infinite terms that are integers.
In that case, it will not be possible for that infinite geometric sequence to contain exactly {{{r}}} terms that are integers.
 
If {{{r=0}}} , regardless of the value of first term {{{a[1]}}} ,
{{{a[2]=a[1]*0=0}}} , and all subsequent terms will be {{{0}}} too.
In that case, there will also be an infinite number of terms that are integers,
and that infinite geometric sequence will not contain exactly {{{r}}} terms that are integers either.