That’s actually a very interesting question. It’s what might be called a “Meta”-question, a question about the questioning process. There are actually logic problems where the correct way to proceed is to ask a meta-question.

For instance…

Suppose you are faced with two brothers, one of whom always lies, and one of whom always tells the truth. Further suppose you have only one allowed question, you can only ask one brother, and you don’t know which is which. What is the proper question to ask in order to get a sure answer as to which door leads to a treasure, and which leads to doom?

Answer that meta-question properly, and you’ll get the treasure.
The right answer is to pick either brother and ask “If I asked your brother, which door would he tell me to open?” (another meta-question). Whatever answer you get, do the opposite. The reason is that you’re effectively routing the question through both brothers this way, which ensures you’ll get one truth and one lie, leaving you with a dependably wrong answer.

Unfortunately, in regards to your query, the question is more interesting than the answer. The current trend in philosophy is to ask a very narrow question, generally about some fine shade of meaning, and to explore the answer at great length. In particular, people have a tendency to seize upon some small weakness of a previous philosopher, and exploit it for the purposes of academic publishing.

However, if you want the questions historically considered by philosophers, or conversely, had you asked, what questions do I think philosophers should ask, the answers would be:

What is the meaning of life? What is the nature of existence? What does it mean to be a good person? What is truth? What is beauty? Toward what ends does the universe strive?