I saw a video talking about this on YouTube. I think given the nature of the show they could rotate out members every season. It's not Arrow or the Flash where there's a title character that has to remain in the cast to keep the show going. Also unlike the other DC properties there really isn't a comic book counterpart of the team. The Legends really only exist in this universe so there isn't any source material to draw from. Which means any character, as long as they're owned by DC comics, can join the team.
Anyway as for characters I'd like to see join the Legends... Booster Gold is a guy from the future that traveled into the past to try to become famous as a super hero. It would be interesting if the way that he actually got to the past was by meeting the Legends in his own time and sneaking on board the wave rider to hitch a ride to the past. Then maybe they catch him, aren't really sure exactly what time period he came from because he's so unimportant to history that Gideon doesn't even have a record of him.
Plus it makes sense for Booster Gold to exist in this universe because of Gideon. We've already seen two versions of Gideon in this universe. The first one in Eobard Thawn's time vault on the Flash. The second being the Wave Rider's computer on Legends. But in the comics the original version of Gideon was a program accessed by Skeets, Booster Gold's robot side kick. It's through Skeets and Gideon that Booster Gold is able to access historical records of crimes before they even happen. So it's also quite possible that the reason Skeets even has access to Gideon is because he made a copy of the computer program while on board the Wave Rider. Or maybe they just completely stole Gideon leaving the Wave Rider without a computer. Booster Gold did steal Skeets and all his other tech from Kord industries.
It also makes sense for Blue Beetle to appear in the series at some point because Kord Industries has already been mentioned a few times on Arrow and the Flash. Mostly whenever someone is stealing tech from there. Though so far we've never seen either version of the Blue Beetle. Ted Kord is the original Blue Beetle.
Some other characters I'd really like to see on Legends of Tomorrow... and this could really help with the Supergirl series existing as Earth 38. (I have no idea why they picked that number for Supergirl's Earth) Rokk Krinn/Comic Boy, Imra Ardeen/Saturn Girl, and Reep Daggle/Chameleon. It would be interesting to see two different versions of these characters portrayed by the same actors in both Supergirl and Legends of Tomorrow. The Earth 38 versions of the characters would be more like the original source material.
The three characters I mentioned are members of the Legion of Superheroes, a team that was formed in the future based on the legacy that Superman left behind. They typically are portrayed wearing matching uniforms just with different chest symbols. Of course we know that Superman doesn't exist on Earth 1 because if you remember when Flash and Supergirl first met Barry had no idea who Superman or Supergirl were suggesting that neither of them exist on his Earth. I think that Earth 1 Kal-El either landed on a different planet, never made if off Krypton before it was destroyed, or was never born. Without the existence of Superman the Legion of Superheroes would never be formed so it's members could potentially join the Legends of Tomorrow instead with more individualized costumes. It'd be interesting to see what versions of these character who have never heard of Superman before would be like. The legion version basically worship Superman like a God with every member having a detailed record of his entire life's history memorized. And they fight crime based on the example he set. The Legends versions could still be heroes just without the Superman obsession. It'd also be interesting if they do crossovers where the Earth 38 versions and the Earth 1 versions of those character actually meet each other and work together because their methods would be very different. So far there's no chances of something like this ever happening because there are no characters that exist in both universes. I'd really like to see some doppelgangers pop up on Earth 1 and Earth 38 but with no actors/characters being shared between the shows it seems like that will never happen.
One of the main reasons I picked these three characters is because there was an episode of Superman the Animated series in which these three characters time travel back to when Clark was a kid to help save him from a time traveling Brainiac who's attempting to kill him before he can ever becomes Superman. Clark is a lot weaker during this time as his powers aren't fully developed yet. So this means there would be a chance for the Legion to appear on Supergirl in a similar way and for them also to appear in Legends of Tomorrow. Both versions would then be time travelers. There are other members of the Legion as well but the three I mentioned are the ones I'm most familiar with so I'd really like to see alternate universe versions of them appear on both shows.
It seems kind of weird that there are seemingly doppelgangers of everyone on Earth 1 on almost every Earth except Earth 38 and there are no doppelgangers of Earth 38 characters on any other Earths. This could help fix that and bridge that gap if they introduce just a hand full of characters that can appear on all four shows.
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