So, I'm trying to figure out how to make my inventory have as many slots as possible.  I have 45, and that INCLUDES the extra one in Zombie Island.  I'm pretty sure I have all of them from the first playthrough, and received one in the second.  How many are possible?  I see videos with people having nearly 70 of them, and want to know if this is possible without hacking or cheating. 
 
Hoping someone out there can help me out.

First off, you're missing one from Playthrough 1.  You should, at the end of Playthrough 1, have 42 inventory slots if you saved every Claptrap.
 
You can get up to 15 more on Playthrough 2 from the first 5 Claptraps, but that's random; sometimes you'll get a grenade mod.  As mentioned, there's one in Zombie Island, which can be obtained once in Playthrough 1 and once in Playthrough 2, and I believe there's one in the General Knoxx DLC, as well.  That's a total of 66 slots, if my math is correct.

I have done every Claptrap that exists and I have 45 slots. 
 
The reason people have more then that is there is a glitch where a random claptrap in play through 2 will give you one.  It's kind of pointless.  Really you only need 2 of each weapon anyway.  And if you have Mad Moxxi there is a bank.  I have about 25 of my backpack slots filled and I got about 6 weapons in my bank.

I found the claptrap in the General Knoxx DLC (Lockdown Palace), but he just gave me a grenade mod.  What a jerk. 
 
And I just like holding onto as many of the orange weapons as I can, as well as a copy of any Eridian weapon I find.  Of course, it doesn't mean that I USE them all that much, so I generally have to learn to let go of thngs that I'm probably not going to use.  I don't want to spend the ten dollars on Mad Moxxi, as I'm not terribly interested in that piece of DLC, and don't want to spend money on just a bank. 
 
Thanks, though.  I have no idea which on I'm missing in Playthrough 1, though.  Could have sworn I had them all, but I guess I'll have to triple check.

Almost positive that I have that one, but I suppose I can look through the massive list of completed quests and juxtapose that with a list of all of the claptraps.

Nah.  You have them all.  I am almost 100% sure I have done every clap trap in the game.  45 slots that's it.  Unless you do the play through 2 glitch.  You're good.  And I don't blame you about Mad Moxxi.  It's fairly mediocre.  I did one of the long Arenas.  I can't remember how many rounds it was.  But it took me and 2 other friends about 4 hours.  After that I have never done another one.

70?  o_O