The River doesn't seem to belong jammed in with two happy musicals, so unless there is some reason for that inclusion, it seems the consensus is Kinky.
See I'd be the voice of dissent and say to see "The River" for one simple reason: The other two shows are on tour at present. I doubt you'll get "The River" where you live (or on tour at all). Hugh Jackman is a rather amazing stage actor to watch.
In the case of Pippin, the tour cast is a lot better than the Broadway one (I mean you have Kyle Dean now, Lucie is supposedly great, and John Rubinstein is fantastic). I did hear that the Kinky Boots tour cast was great as well.
Can’t speak to “The River,” but between price and many reports of a lacking script, I would count that one out. I’d say “Pippin.” Especially since it’s closing soon and “Kinky” will be around for a while – I would actually think seeing “Kinky” without Billy Porter in the future would be a benefit to not seeing it now, as I was not a big fan of the show and disliked Porter's performance even more than the show itself.
I was going to say the same exact thing as Dreaming. You will never be able to see The River again, at least with Hugh Jackman. Not the case with the other two....they will be touring for a long time, Kinky in particular. (And that new Pippin seriously does not look like any Pippin I have ever seen...they are usually 'boyishly cute'...he may be a great singer (I have never seen the Voice), but he looks creepy).
I loved both Kinky Boots and Pippin, saw them both twice, and plan to see them on the road again (I live in Fla and only get to NYC two or three times a year...so I won't actually see The River, which is disappointing).
Finally, I did see Billy Porter the first time, and his understudy the second. While his understudy was not as good as he was, it did not really detract from my enjoying the show...I still loved it.