Once you get the Atlas V1 pass which is given to you pretty early on in the Atlas path you can access the second door in the spaceports which is home to a back pack slot upgrade store as well as usually a free multitool/suit upgrade.
Today I did a little ship farming like Elkin mentioned and I'm now up to 45 slots

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Still having a bitch of a time getting to the center though I'm basically just using black holes over and over again up until my ship is completely crippled then I'm having to farm on a planet for a few hours for repairs and warp cores before repeating the same process over and over. Still 143911.8 light years from the center of the universe so it will probably be another two to three weeks before i get there if I don't move on to another game before then

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Curious if anyone has managed to get there yet seems nearly impossible but I'm sure there is someone who has been playing 24/7 since launch who is close if not there already.
Edit: Black holes took out 3 warp drives in a row so I'm calling it a night since that will be another 2-5 hours to repair them all again

. They really need a repair garage or some shielding system upgrade that you can install for black holes having to collect materials to repair this crappy damage which is only there to drag out the game is annoying as hell. Especially given the amount of progress you are rewarded for your effort. Really having to take 200+ black holes to get to the center of the universe is bullshit especially when each and every black hole knocks out a piece of your equipment making things even more tedious.