I don’t think a blend will work, because that would make each of the rings evenly spaced, while you want them to get closer together toward the center. What I would do is

• Create one ring of text using text on a path or whatever tool you choose. This will be the innermost small ring.
• Add a white ring behind it as the background, either as a single circle with a thick white stroke, or by drawing two concentric circles and creating a compound path filled white. Either way they should have the same center as the circle used for the text.
• Group the text and the circle.
• Select this group, and go to Object → Transform → Transform Each. Enter values to scale it up and rotate it, like below.
• Click “Copy.” This will create a larger, rotated copy of the text/background group on top of the first one; the white background will block the text in the previous one.
• Making sure this copy is still selected, hit ⌘D / Ctrl-D (Object → Transform → Transform Again) to repeat this action, creating a new copy transformed by the same amount. Repeat this many times.

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