Learn English – Is “With which” correct

I don't believe that "for many times" is grammatical in any variety of English.

^{(Well, unless you contrive an example where "times" is the plural of "a given time of occurrence" and there's an elided "of the": "Class times are listed below. For many times, you can click the link for more information.")}

You can use "for" with an ordinal: for the first time, for the 20th time, for the last time. Notice that it's always time, not times, in these examples. You can also use "for" with a duration: for 100 years, for the summer, for a long time.

"Many times" behaves the same way as "100 times" or "just a few times", and adding a "for" is not grammatical.

Mark has been to Hong Kong many times.
Mark has been in Hong Kong for two years now.

After Jessica failed 10000 times, she finally created a working robot.
After Jessica failed for the 10000th time, she tried a different approach.

I disagree with Ekn.

"For the first few days" implies that something was going on continuously for days. While technically you probably thought about other things during those first few days, like eating and going to the bathroom, it is metaphorically valid, and certainly better in a literary since, to say that it was continuously ongoing. I would absolutely vote in favor of using "for" here.

However, "in" is not invalid. It says to me that at any time in those first few days that you attempted to figure it out, you couldn't, but not necessarily that you did so a lot for those first few days.

So both are valid, but in the context of a breakup I would use "for".