The only one I know of is the iPhone. I think it's safe to assume that one of the reasons Verizon turned down the original iPhone was that they didn't like that they wouldn't have much control over it if at all. One of Apple's conditions certainly would have been no Verizon logo anywhere on the device and possibly none of the V-Cast / VZ Navigator type preloaded apps (and I'm sure the original $600 price tag had at least a little bit to do with it, haha). But over time Verizon recognized that many of their customers wanted an iPhone and many people were switching to AT&T since they couldn't get one on Verizon, so they eventually gave in to some of Apple's demands in order to carry the iPhone.
Google originally planned to have a Nexus One on Verizon, but the rumored reasons why it didn't end up happening was that Verizon and Google were in disputes possibly over how they would sell the device, since Google wanted to sell the device themselves online only, and Verizon wanted to sell it in their retail stores. Sound familiar?
The question is does Verizon feel like they need Google and their new flagship phone right now, like they did with the iPhone? I'm sure Verizon thinks they don't, however I think that just like the iPhone, many Verizon customers right now, although obviously not as many as with the iPhone, have been waiting for a long time for a "stock" Android phone considering the last Verizon phone to have stock Android was original Droid. I don't think Verizon necessarily "needs" Google this time like they did with the original Droid or with the iPhone, they are selling tons of Android phones without a "stock" / "pure Google" phone in the line-up, but I think they have shown some interest in staying strong partners with Google and Android, and they certainly like being first to launch with Google's newest version of Android (first with 2.0, first with 3.0, and now 4.0?). I also think Google has lately been showing an interest in Verizon (XOOM for example) especially due to Verizon's LTE network maturing and expanding throughout the country rather quickly.
It's possible that Google has been waiting for Verizon to get LTE up and running to bring their flagship device back to Verizon. It didn't occur to me until recently that when the Nexus S was announced/released Verizon's LTE network was brand new (officially launched on Dec 5th) but "wasn't ready" for phones yet (just Mi-Fis and USB modems). If the Nexus S, released in December, had come to Verizon it most likely wouldn't have had LTE and that would have been seen as a huge missed opportunity just like how some people felt about it not being dual-core after all the initial rumors that it was.
Most of this talk is pointless though as it probably sounds like I'm questioning whether the device will come to Verizon, and I think it's pretty clear that this "first device to run Ice Cream Sandwich," whether or not it is branded as a "Nexus" device, is now pretty much certain be on Verizon and with LTE. The only reason I mention all of this is because I'm curious to see how "exclusive" this device will be for Verizon. The exclusivity of carrier is mostly dependent on how much Verizon really wants this device, and how exclusive Google will let them be. It could simply be the exclusive "launch partner" or something where the GSM or Sprint WiMAX version of the exact same phone is released a month or so later, or it could be a similar situation that happened with the original Droid and Nexus One, where they released two completely different "flagship" phones 2 months apart. Although I'd expect it to be either the exact same device with different radios, or very similar maybe slightly different devices like the difference between Galaxy S II variants, possibly one with a keyboard and one without as the rumors say.
I know, I know, "TL;DR." I don't blame you.