Well done, traveller, you've almost made it to the end of survival week. And you're still capable of rational thought and reading! Truly you are an immortal of our time. I've got a reward for you. Primitives, a student game by seven final year-ers at the Queensland University of Technology, starts in a similar fashion - chop down a tree, make a campfire, get some stone, try not to eat your own face off in hunger. The twist comes when you start building huts, causing AI compatriots to arrive and turn the game into more of a third-person RTS. It's also totally free and available in a beta state now. Trailer and some thoughts below.
It's an interesting prototype and idea, but is a fair bit of work short of a finished game. Developers Ugly Stick are still working away, planning to release updates publicly. What's currently there is functional but buggy and cosmetically very basic - poor animation and unfinished textures. It's playable but it'll take a few goes before you reach the tribe stage rather than dying of starvation with only a few rocks to keep you warm.
My first playthrough ended after I punched a wolf to death then a friend of his managed to eat me. In a wonderful display of a not-quite-finished physics engine, my character model then rocketed into the sky, ragdolling wildly. My second I strayed too far from camp at night and froze to death after killing a mammoth with a spear. It's definitely a survival game.