Sharp cheddar and Cream.
It was hard to pick Cream over Gouda and Edam and there are at least five I would have in my favorites list. There aren’t many cheeses that I actually dislike.
Sharp Cheddar and Parmesan. Cheddar is absolutely non-negotiable, and by far the cheese that I eat the most of. The second was between Parm and Mozzarella, and it was pretty close, but the only thing I really have a ton of mozz on is pizza, and I’ve had some pretty decent pizzas with cheddar. I have parm on a bunch of different things. So that’s that.
Agreed that this is a terrible, traumatic question.
Mozzarella & Colby
I started by choosing all the cheeses I couldn’t live without - I figured that I would then start deleting until I ended up with two. Unfortunately I was unable to delete any so I defied the cheese gods and left them all in!
Cheddar–generally useful for everything.
Cougar Gold–heaven in cheese form. http://www.wsu.edu/creamery/cheesetour.htm
I chose sharp cheddar and asiago, but your answer is better.
Ow. This was a real Sophie’s Choice of a poll.
Only two cheeses? That’s just mean.
Anyway, sharp cheddar and, well, stilton was my instinctive other choice, but then I thought,“if I don’t say cream cheese, and this actually came true my morning bagels would much the poorer”, so there you have it.
Can’t say I’d be over the moon about having my parmesan taken away from me, either.
Two cheeses! That’s not a choice, that’s a penance!
Voting for sharp cheddar and Swiss because those are the ones I eat the most often. But a world without even the occasional taste of brie . . . 
Sharp cheddar (the sharper the better) and bleu.
Here with the others to decry how impossible it would be to only live with two cheeses. This, coming from a woman who at the culmination of a job interview, instructed the potential boss on how to smuggle cheese in from the U.K. where he was headed (I brought back almost seven pounds during my last trip).
I wish to have my ashes scattered between Borough Market and the Harrods food basement. Or Neal’s Yard. So many choices…
I have told my friends to kill me if I become lactose-intolerant as I will have no reason to live without my beloved cheese.
Bucheron with Sauternes is a favored dessert.
I can recommend Parrano with a Pedro Ximinez sherry.
Port and 5-year Gouda works well too.
I’ve been known to lick Epoisses off a willing partner. <sigh>
The Wallace and Gromit Wensleydale episode practically saved the dying dairy.
Stinking Bishop is stunning.
Ah, the joys of a cheese…
Not a problem. You’ll still be able to enjoy aged cheeses. No lactose left in them.
It would be necessary to storm the Cheese Heavens and free all of the wonderous varieties for the benefit of mankind. I refuse to live with only two varieties of cheese.
I love you for this.
I’m very amused and gratified by how difficult and cruel you’ve all found the Cheese Gods’ decree to be! I figured you would… And if it was easy it wouldn’t be any fun.
I’m surprised by Gouda’s strong support…
I voted Sharp Cheddar, but I’d want a really mature vintage cheddar - one that’s almost tangy enough to hurt, and has those little crunchy calcified whey granules in it.
I also voted ‘Other Blue’ Stilton would be fine, or another cheese of that type - as long as it’s abundantly blue, with a strong flavour.
Sharp Cheddar and Parmesan.
Interesting to find them back to back in the list selection.
I actually had to pick up some Parmesan today. I find myself putting it on next to everything. My diet is big on cheap Itallian [Pizza Buffets, Spaghetti] and I often find that Parmesan compliments a dish well, OR there are times when I wish I could shake some on. [Hence I finally remembered to get some today.] Actually found a recipe once that was nothing but Parmesan heated at 400 for some length of time.
Cheddar. What can I say about Cheddar that hasn’t already been said? [And looking at the results, this seems to be even more true than I thought.] There is a reason why [With the possible exception of ““American”” cheese (what ever that is) ] Cheddar is the default go-to cheese. It seems to be more versatile than the others. Like “Apple” is for fruit, so Cheddar is for Cheese. There is some profound truth there. As Fallon compared Tuna Casserole to Sweatpants, I must conclude that Cheddar is as casual. If I am to be locked into a cheese, I’d want sweatpants and flip flops. The Sharp aspect would need to be on the table [heh] in order to even approach the OP question. Mild would get old fast.
A third cheese would probably be Pepper Jack, if I could have had a third. Growing up, as a family we didn’t do spicy foods all that much. In recent memory, I have tested my boundaries on spicy, and have moved up to medium. Same for Pepper Jack. I like the… tingle, the for lack of better words, the challenge that comes with spicy foods. I don’t like to take it on a daily basis, let alone to be locked into it for “the rest of your life”.
Colby Jack would be in the running as well. But I guess it kinda cheats the question at hand. [Aside from not being an option.] I would argue for it with the same statements I gave for Sharp Cheddar.
Poor unloved Stilton – well, apparently well loved by the posters but cruelly neglected by the OP. Shame on you Stoid. 
Really surprised that cream cheese isn’t fairing better,since a life without cheesecake is no life at all. I use it in so many things, I had to vote Sharp Cheddar and Cream. But really,life is not worth living without at least 6 varieties of cheese. 10. Maybe…14.