Turing completeness of Bitcoin Script
If scripts were Turing-complete, you could construct a fairly short script that took an extremely long time to run (a la the Busy Beaver) or contained an infinite loop.  This would tend to result in a denial of service against everyone on the network, when they tried to verify the transaction.  

And there would be no general way to tell whether a script would loop infinitely or would eventually finish: that's the halting problem.

One could try to avoid this by including some sort of cutoff on how long a client will spend verifying a transaction, but it would have to be done very carefully so as to be consistent across all clients: if some accept it and others don't, the block chain will fork. 

It seems simpler to avoid the situation all together.