Suppose I have 2 wallets and 1 bitcoin and I write a program to make transfers of 1 satoshi from one wallet into the other.
Now suppose I have 100 bitcoins. Wouldn't that saturate the network?
Your transactions would be given the lowest priority as they are not paying fees, also if anyone is actually marginally successful in spamming the system it is entirely possible that the larger pools will simply not take their transactions, in fact some pools do filter out some transactions. This thread shows some of the restrictions that some mining pools already have.