こんにちは (Konnichiwa) Jyuune and JohnH197,
Thank you for your question!
There are two different grammar points being used in this sentence: 〜てもいい (~te mo ī) “may” and 〜てもらう (~te morau) “to do (for me)." Let's break them up and take a closer look at how they work.
First, let's look at 〜てもいい (~te mo ī).
This is a structure used with the 〜て form of a verb to give permission. It means “may” or “can.” For example:
食べてもいいです。 (Tabete mo ī desu.) “(You) may eat.”
写真を撮ってもいいです。 (Shashin o totte mo ī desu.) “(You) may take a photo.”
By adding か (ka) after 写真を撮ってもいいです (shashin o totte mo ī desu), we can ask for permission:
写真を撮ってもいいですか？ (Shashin o totte mo ī desu ka?) “May (I) take a photo?"
You can read more about how to use this grammar point in this lesson:
Next, let's look at 〜てもらう (~te morau).
By itself, the verb もらう (morau) or もらいます (moraimasu) means “to receive.”
When we use it with the 〜て (~te) form of a verb, we can ask for a favor. 〜てもらう (~Te morau) literally means “to receive the doing of (something)."
書いてもらう (kaite morau) “to write it (for me)"
写真を撮ってもらう (shashin o totte morau) “to take a picture (for me)”
I have been informed by the Japanese team that an in-depth explanation of this grammar point will be covered in the new level 3 course when it is released.
Now let's try putting 〜てもいい (~te mo ī) and 〜てもらう (~te morau) together:
写真を撮ってもらってもいいですか？ (Shashin o totte moratte mo ī desu ka?) “Will (you) take a picture for me?”
What this sentence is literally saying is “May (I) receive the doing of taking a photo.” In other words, “Will you take a picture for me?"
The reason there are two 〜て (~te) forms in this sentence is because the first one is part of the structure 〜てもらう (~te morau), and the second one is part of the structure 〜てもいい (~te mo ī).
I hope this helped!
Please let me know if you have any other questions.
勉強を頑張ってください！ (Benkyō o ganbatte kudasai!)