Learn English – Been to America three times/for three times

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.

Family will be there for you at the most difficult times no matter what.

In phrases referencing time, at would be the best choice. But in this example, times can refer to time or events.

You might use in if you mean events:

Family will be there for you in the most difficult situations no matter what.

However, in your original phrase, during would be a better preposition to use, since a time span is implied.

Family will be there for you during the most difficult times no matter what.