Not an expert, but after reading around OP is saying you only need a constant sized buffer (that could be as little as a character deep) to read the whole file

Not an expert, but after reading around OP is saying you only need a constant sized buffer (that could be as little as a character deep) to read the whole file looking for line breaks. This roughly takes O(1) space (A lot of commenters are confusing time vs space requirements) but there's a little nitpick to be found, in extremely big files the line counter itself would be O(log(n)) and would dominate.