Context: I walked a dog three times today.
“I was walking a dog for the third times today.”
“I was walking a dog for three times today.”
Are both valid expressions?
I walked a dog three times today.
means that you walked the dog three times, and all three walks happened today.
I walked a dog for the third time today.
means that you walked the dog three times overall, and the third walk happened today. The first and the second walk may have happened today as well, or they may have happened at an earlier time.
Please note that "for the third times" is wrong. This expression only refers to one walk, namely the third one, so you need to use the singular: "For the third time".
"For three times" maybe not technically wrong (I'm not really sure about that part), but at least it feels off.