Why do we need אף פעם then?
The two beautiful and expressive words we mentioned before are.. well, expressive! They are not really neutral!
If you say
!מֵעוֹלָם לֹא אָכַלְתִּי עוּגָה כֹּל כָּךְ טְעִימָה
It's like saying "I have never ever in my life eaten such a tasty cake!" (maybe even with several exclamation marks! :) ) rather than just saying "I never ate such a tasty cake before."
So - how do we tone it down?
We just use the simple "never" word instead - אַף פַּעַם.
אַף פַּעַם לֹא אָכַלְתִּי עוּגָה עַל כָּךְ טְעִימָה.
Examples with מעולם and לעולם:
מֵעוֹלָם רַגְלוֹ שֶׁל אָדָם לֹא דָּרְכָה עַל הָאֲדָמוֹת הַנְּטוּשׁוֹת הָאֵלֶּה. - "Never has a human foot trodden on these abandon lands."
וְהִנֵּה הַקְּלָלָה שֶׁלִּי - לָעוֹלָם לֹא תּוּכְלִי לִצְחֹק יוֹתֵר! "And here is my curse - you'll never be able to laugh any more!"
You see? In these two examples the whole style is more elevated and the situation is not very regular!
Whereas, if we speak about everyday matters and you mean no emphasis on the "never" word, - we'd definitely choose to use אַף פַּעַם.
Examples with אף פעם:
...מִי זֶה? אַף פַּעַם לֹא רָאִיתִי אוֹתוֹ קֹדֶם Who is this? I've never seen him before.
?לָמָּה אַתָּה אַף פַּעַם לֹא מְתַקְשֵׁר Why do you never call?
-------last point-------
All well and clear now (hopefully), but there is still one question left - I hope you wanted to ask it!
Why do I write מעולם\לעולם\אף פעם but the dictionary adds a "no" to it -
מֵעוֹלָם לֹא \ לְעוֹלָם לֹא \ אַף פַּעַם לֹא?
Well, how do you say in Hebrew: "It's never late to learn a language!"?
Right, אַף פַּעַם לֹא מְאֻחָר לִלְמֹד שֶׁפֶּה! ;)
Do you see the difference? It's as if we say "it's never NOT late!"
And it's typical for Hebrew - it always uses the "no" word in addition to a negative word (like "never," "no one," "nothing," etc.)
Examples:
אַף אֶחָד לֹא נִמְצָא - No one is present
שׁוּם דָּבָר לֹא מוּבָן - Nothing is clear
שׁוּם עָצָה לֹא תַּעֲזֹר - No advice will help
You see? Whereas in English one "no" is enough, in Hebrew we use the so-called "double negation" - negative word plus "no"! ;)