I have looked through the relevant gemoros in Menachos and Krisos as well as the Rambam in H' Klei Hamikdosh but I can't find a source for what quality of olive oil was used for the shemen hamishchah. Was it "kasis" or not? Was it okay to bring any of the nine levels mentioned in Menachos (86a-b)? Thanks.
Partial/tentative answer for now:
Rashi's commentary to Shemot 27:20 and 29:40, which specify the oil for the Menora and for the meal-offerings, respectively, cites inferences from these verses made in Menachot 86a-b to the effect that:
Katit ("crushed" - first drip) oil was only required for the Menora, since 27:20 says "כָּתִית לַמָּאוֹר" - "crushed for lighting."
Katit oil was also allowed, but not required for the meal offerings, since 29:40 says "כָּתִית" - "crushed," but doesn't further specify something like "for meal offerings."
I see no reason that the first exclusion wouldn't also apply to the anointing oil, meaning that like meal offerings, it also wouldn't require katit oil. The inference there is that this requirement only applied for Menora oil. (Unless there's some verse about anointing oil that I'm not aware of that sets it apart as well.)
I can't tell, from these inferences, whether katit oil was allowed for anointing. From the second inference, which allows katit oil for meal offerings, it doesn't necessarily follow that it was allowed for other purposes.
Shemot 30:24, in the commandment to make the anointing oil, does not mention "katit."
I used this question as the basis of a recent Substack post, where I came up with my own answer, and compared it with the answer generated by ChatGPT.
In short, I would say that there is no requirement that the anointing oil would require katit level quality. I have a few reasons for this.
Firstly, from a logical perspective. The quality of the oil (say, extra-virgin olive oil) is a qualitative difference that can be sensed in the clarity of the light produced. Similarly, from a culinary perspective, it is something that might be tasted, so the Mishnah had to consider it (via kal vachomer) and then reject it (via a diyuk). For pouring oil on someone's head or anointing vessels, should this really matter?
Second, an argument from silence. Usually I would say that absence of evidence is not evidence of absence. But here, the Mishnah and Gemara explicitly consider other oil situations. And if this were a requirement, we would expect a Biblical verse, Mishnah, Tosefta, Midrash Halacha, or Rishon to say so.
Thirdly, from a hermeneutical perspective. The Mishnah presents a kal vachomer. If for mere light it is required, then certainly for something that goes on the altar! And the rejection is that the pasuk explicitly says לַמָּא֑וֹר, that only for light. Absent an explicit Biblical verse (see point 2), the only reason we would think it would be required was by analogy or kal vachomer. But we already have the diyuk of לַמָּא֑וֹר to reject such a kal vachomer. (And perhaps the idea behind this diyuk is what I wrote in point 1).
It says pure beaten in Shemot 27:20
'And thou--thou dost command the sons of Israel, and they bring unto thee pure beaten olive oil for the light, to cause the lamp to go up continually;
Even though it talks about Menorah, it is the same oil that was donated(Shemot 25:6), so I think this would be the answer.