Upgrading from 1.6.7 to 1.6.10, something along the way broke the costs for claiming hexes. I can claim hexes despite not having enough coins in my inventory (and no free hex claims). I also don't have a piggy bank, whatever that is (from reading changelog).

I can provide my save file if necessary.

The update which introduced the Piggy Bank feature, along with some of the updates after it, made it possible to "lose" coins upon spending any coins anywhere at all. But what was happening was the coins were going into your piggy bank secretly before it was unlocked. Did you migrate your save through those updates and unknowingly put coins into your piggy bank? If you unlock the Piggy Bank from the "Industrial Garden" quest on Gleba and see more coins in it than what you had before completing it, that would be the reason for your free claims.

Can quickly verify this by running the command /complete-quest industrial-garden and check the piggy bank (custom GUI above your inventory). It will put coins in there for you automatically on quest completion, so maybe empty your inventory of coins first before checking. Then you can revert to your non-cheated save copy.

Empty piggy bank. I've never been to Gleba this save.

I've been playing since I originally posted, and tried the version upgrade again now. Can't seem to reproduce the issue.

Then I will need your save. Can you host it as a multiplayer game so I can join you and download the save?

In case it's useful:
https://drive.google.com/file/d/1OIfgPs8T2NGW97bM3A_brR9npS_d-J8z/view?usp=sharing

Mod list:
base 2.0.73
AutoDeconstruct 1.0.10
elevated-rails 2.0.73
flib 0.16.5
quality 2.0.73
Milestones 1.4.7
simultaneous-research 0.7.4
space-age 2.0.73
hextorio 1.6.10 (previously 1.6.7)

I have a reproducible save, I think. This may or may not be the same as the original bug I encountered, but there's definitely something weird with hex costs.
https://drive.google.com/file/d/1xvGAWESP_2lsv4_jLPMAYcPYuZRPQVPN/view?usp=sharing

The tile to my immediate west appears to cost 1 purple, 60k yellow coins, as do the tiles to my immediate southwest and southeast.
Buy the tile to my southwest. Now looking at the tile to my west, it still appears to cost 1 purple, 60k yellow.
Now looking at the tile to my southeast now shows a cost of 6 purple, 40k yellow coins. Without buying anything, looking at the tile to my west now shows 6 purple, 40k yellow coins as well.

For reference, my starting tile on Vulcanus is the one my purple coin provider chest is in (Bank train station).

Claim costs are now calculated based on "valid path" distances from spawn, a very recent change (I think 1.6.8). I did accidentally forget to migrate the claim costs of existing hexes on saves, so it's only newly generated hexes that have the new claim cost formula. You can make it recalculate all hex costs by tweaking claim cost settings, though. Just set it to something and set it right back. Then, all costs will be consistent again. That is if I'm understanding your issue correctly. That's the most likely cause of the discrepancy. But if that doesn't fix it, I'll look at the save and see what new thing is really happening.

I tried changing the planet claim cost multipliers (mod settings, map tab) from the recommended values at .2 BTE to all 1, confirmed the settings changed, ran the game a bit, then changed them back to the .2 BTE recommended values. That should force the recalculating as you say, but there's definitely something broken somewhere.

Try this save file, made after the above settings changes. I have no coins or free claims. I see some hexes with old prices, but some with a new price of 8 purple 9999 yellow, and yet claiming them succeeds. Claiming outward in an arbitrary direction, I find new hexes with the same 8 purple 9999 yellow price that are somehow also claimable, seemingly infinitely.
https://drive.google.com/file/d/1maCQVT21zLRT-8kiAgYsS6LLr48VDldw/view?usp=sharing

I'm seeing your bug now. Your save will be migrated to have corrected hex claim prices in 1.7.0!

Excellent. Glad I was able to help.

There's still some issue with the hex claiming, on 1.7.2

I'm on gleba, no coins in inventory, no piggy bank still, no free gleba claims.
Trying to claim the hex to my east (cost 3240) fails, as expected. But if you take one purple coin from the box, put it in your inventory, and then try to claim, it will succeed when it should've still been insufficient.

https://drive.google.com/file/d/1kBRWy-vhTpbJNVW4ix21cZflVQYWzc0W/view?usp=sharing

It seems i can place landfill freely and build in chunks i have not claimed. Those 2 are essentially also free claiming and a bug?

It's intentionally allowed.