Anderson Cooper would be good, I think. Serious, but not too much, and he seems to have the ability to put his guests (contestants, on Jeopardy) at ease.
Andy Richter.
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The problem with Anderson Cooper is that he’s a true straight man. He works with guests who are professional performers for the most part.
Alex Trebek had a good sense of when he had a contestant who was a bit of a ham, and would play “straight man,” and let that contestant perform, and when someone needed Trebek to do the performing. Also, occasionally someone thought they could perform, but they actually were bombing, and Trebek had to squelch them. He always did it elegantly. The trick is finding someone who can do all those things.
What’s Mayim Bialik doing now that Big Bang Theory is over?
This: Call Me Kat - Wikipedia
It looks ghastly.
I think that Anderson Cooper is a fairly unlikely candidate, since already he makes about 3/4ths of Trebek’s salary at CNN, and even assuming he had the time for two jobs, working on a game show would tend to hurt his credibility as a news anchor.
Perhaps if he was tired of working at CNN and wanted something easier, but I don’t get the sense that that’s the case. Also, the optics of quitting the news for a game show would be odd and open him up to some ridicule, IMHO.
The same goes for several other putative candidates who already have good jobs.
I have no opinion on a new host, but I really hope they use the revamping as an opportunity to get rid of the awful “tell us something interesting about yourself” interlude.
I can guarantee you they will not get rid of that, or any other key element of the show. There are simply too many rabid fans to mess around with any significant aspect of the 37-year old format. Maybe a new set.
A new host will be a traumatic enough change.
They did modify the set after COVID to separate the three contestant podiums. I think Alex said something about how they liked the change and might keep it even when social distancing is no longer needed.
ISWYDT. 
Yet another vote for LeVar Burton.
Oh, hush.
Definitely LeVar.
That and the Clue Crew / talking celebrity crap.
I’d like to see it offered to Steve Harvey, Drew Carey or Wayne Brady but they would all probably prefer something more laughy. I’d also welcome Anderson Cooper doing something different than his CNN gig. Or maybe Erin Burnett. Trebek was so good because he was like the schoolmaster that knows everything and could pronounce all the foreign words and phrases; they might have to water down some of the subject matter to accommodate the new host.
Buzzy Cohen or Brad Rutter. I swear I’ll stop watching if Ken Jennings gets the gig.
Ugh no. I love Wayne on LMAD and Drew on PRIS. But they aren’t really the right fit for the show. The important thing about the Jeopardy host is that they sink into the background as it were. The show isn’t about them.
LeVar Burton would be terrific. Wish I’d have thought of him first. A real long shot, but another interesting candidate would be Jon Stewart. Though I think he’s pretty happy being retired in NJ and isn’t relocating to LA.
I really don’t think they’ll go with a former contestant, being a good host is a surprising difficult skill to master and I just don’t think an amateur can swing it. Wouldn’t mind seeing Ken Jennings get a temporary shot at filling in just to see how it goes, but I would not expect them to get it long term.
Also, people suggesting Betty White and other similarly old people must be out of their damn minds. Jeopardy is not a folksy type job. You need to be extremely articulate and precise with your language and really crisp. Also, they aren’t going to put in a replacement who’s likely to die or retire inside of a decade.
Lastly, they aren’t likely to pick a working actor/newsman either. Think of Drew Carey, while he had a really popular show it was over and he wasn’t really doing anything else when they called for TPIR. This was his new career, not a sideline. Jim Parsons, Anderson Cooper, Dan Patrick…those guys aren’t quitting their day jobs for this, and you can’t really do a daily game show like this in your spare time.
What’s Kelsey Grammer doing these days? The big thing against him would be his age (he’s only a month younger than I am).
I think Alex himself suggested Betty White as a joke.
Jim Parsons, Anderson Cooper, Dan Patrick…those guys aren’t quitting their day jobs for this, and you can’t really do a daily game show like this in your spare time.
But you’re mistaken about the bolded part. They tape two days (ten shows) every other week, with a total of 46 taping days per year. Most likely hosts could easily work that schedule around another full time job.