"At least" is an expression of speech signifying, "though something else may be the case, thankfully this is not the case."
"At the least" carries the more literal meaning, "if not more, certainly this much."
The second connotation is also the technically syntactically correct one.
the 가 붙으면 문자적인 의미의 '최저치'를 의미하고
the 가 없는 at least 는 '적어도 이정도는' 이라는 의미가 된다
The meaning of all the three phrases is more or less the same but they have specific usage to express different meanings: as:-
At least: If nothing else/more exactly - The car was damaged, but at least no one was hurt. 적어도 이정도는
At the least: Not less than - used with a particular amount of measure - : At the least, the room is 30 feet long. 그 이상
At the very least: Not less than - The repair will cost $100 at the very least. 최소한도