申し上げる

[Jason]

Could someone please explain 申し上げる in some detail? We touched on it slightly in class when we studied keigo, but not in any real detail. I understand it when I hear it. But I don't understand it well enough to really know why it's used when. Example:

今後ともよろしくお願い申し上げます。

[Jonas]

Hard topic you've chosen! hehe. Here is my understanding of the subject. When you lower yourself compared to the subject of the sentence, you use 申し上げる.

cf.:

理由は課長に申し上げたとおりです

or like in your example, when you are polite towards someone you are speaking directly to.

When the subject isn't something that should be "honored", you use 申します
cf.:

このあたりは昔の地名で申しますと、相模の国でございます。

Here 地名 is the subject, so you would not use 申し上げる in this case

[Jason]

Hard topic you've chosen!

That's the fun part. Thanks for the explanation.

Just from hearing it, I get the sense that it's mega-polite, even among keigo. So, how do you choose when to use:

よろしくお願いいたします。
-versus-
よろしくお願い申し上げます。

[kinoko]

The second sentence seems weird to me. "Please call me well"?

[Bueller_007]

The second sentence seems weird to me. "Please call me well"?

Not weird at all. It's used all the time.
Here, 申し上げる doesn't mean "call".
お願い申し上げます is an expression meaning "please", or I guess, more literally, "I beg of you." It's extremely polite.

I would say, although I'm not sure, that the following list is in order of increasing politeness:
よろしく。
よろしくお願いします。
よろしくお願いいたします。
よろしくお願い申し上げます。

I've used it for my Japanese new year's cards, but I haven't heard it spoken, so maybe it's only used for written Japanese? I'm not sure about that, though.

[Jason]

To memory, I've heard it spoken at least once in a game by a woman who's an attendant to a prince. Seems I've heard it other places too, but I don't remember where now.