how to do the code for repeated addition of a number by 1?
int numberofthings =0;
while (true)
{
numberofthings++;
}
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??????????? make your question more clearly.
for(uint16_t cnt = 0; cnt<65535; cnt++){
}
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Up to infinity is impossible. With integral types you'll run into an overflow, with floating point types adding 1.0 will stop incrementing at some high value.
Without infinity the answer is simple:
x++;
will increment x.
Clearly infinity is impossible, but for all practical purposes, you can create a sum (easily) that's only limited by the amount of available RAM.
You could do it in Flash, but you'd exceed the write limit before you got close to the maximum number 
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Depends, do you want to work smart or hard?
If you know how far that you want to go, you can convert the series into an equation.
for 1-100:
1+100 = 101
2+99 = 101
So, each pair = 101 and you have 100/2 or 50 pairs.
The sum of all is 101 * 50 or 5050.
As far as adding to infinity, that is ill advised, unless you're Buzz Lightyear
as a prelimary exercise
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Will you still be alive when it gets to infinity?
Until the parallel lines intersect
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If you are, just add 1
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