2 I know a man in Christ who was caught up into the third heaven 14 years ago. Whether he was in the body or out of the body, I don't know; God knows. 3 I know that this man—whether in the body or out of the body I do not know, God knows— 4 was caught up into paradise. He heard inexpressible words, which a man is not allowed to speak....Much mental effort has been expended trying to figure out how many levels of heaven there are, and what those levels are.
But the more I think about it, the more I suspect that we make a mistake to approach this issue from a Western mindset. I don't believe Paul is telling us that he's been to Level 3 of 7 levels, or 3 of 28, or whatever.
If you search Biblegateway for the term "heaven", and try to figure out what the terms mean as you look at each reference found, you'll find there's a lot of crossover between usages. Sometimes the heavens are eternal; sometimes they're temporary. Sometimes God dwells in the highest heaven; sometimes He dwells above the highest heaven. Sometimes the stars are in heaven, sometimes rain comes from heaven. In other words, there's no Western-thought-type consistency in what/where heaven is, and we damage the text to try and force-fit an Eastern text into a Western mindset.
Instead, ask yourself what the number three meant to an Eastern mind of the time.
The number 1 usually referred to God ("God is One"), and the number 2 usually referred to humanity (Adam and Eve), so the number 3 referred to the union of God and Humanity (1 + 2 = 3).
From another angle, animals had a body of flesh (1 - Gen 6:17), and the breath of life (2 - Gen 6:17), just as humans did. But humans also had the image of God (3 - Gen 1:26), so again, 3 refers to the union of the worldly image with God's image.
When Paul writes of the third heaven, it seems to me that he's saying he went to a place beyond the two heavens where the birds fly and the stars dwell; he went to the place where God receives human visitors. He went to God's guest parlor, so to speak.
I'm no expert in Eastern thinking, by any stretch. But I seriously doubt that Paul's reference is correctly approached from a Western way of thinking.
I may be wrong.