1. http://www.earlham.edu/~peters/courses/logsys/turing2.htm (This proof looks cool. But I think in step (3) it should be $n+d$ rather than $n+1$.)

  2. http://cis.csuohio.edu/~somos/beaver.ps (This proof not only proves $\Sigma (n)$, i.e., busy beaver function, is uncomputable, but also shows that $\Sigma (n)$ grows faster than any computable function asymptotically.)

  3. I think up of a proof using the recursion theorem. Suppose there is a TM $B$ that can compute the busy beaver function, then construct a TM $M$, starting on blank tape, do the following operation:

* Get its own description $<M>$, via the recursion theorem
* Once you know $<M>$, you know the number of states of $M$. Denote it by $k$. Use $B$ to compute the busy beaver function of $k$
* Write $k+1$ 1's on the tape, leading to a contradiction