These non-palindromics numbers are called "Lychrel Numbers", although no number (in base 10 anyway) can be proven to be ever non-palindromic; they're just non-palindromic in the first few, ah, million iterations. Of course, they also get less likely to become a Lychrel Number after each iteration*.
As of August 14, 2002, someone has calculated 196's iterations up to 41 Million digits; that's around a hundred million iterations.
*: According to Dan Hoey, the probability can be calculated as such:
You suggested that 196 will yield eventually. If you mean it will become a palindrome, I don't think that is very likely. The reason is that the number will only yield a palindrome if there is no carry out of any digit when the addition is performed. If the digits are randomly distributed, the probability of this is about 2^(-n/2) for an n-digit number. The number of digits is about 2/5 of the number of steps taken, and will certainly increase at least once every five steps. So the probability that the number will eventually reach a palindrome on the k'th step should be less than 2^(-k/10), and the sum of that for k=3,000,000 to infinity is something like 10^-90,000.
There are some numbers in other bases for which the process can be shown not to terminate (http://www.seanet.com/~ksbrown/kmath004.htm, for example) but that is because a particularly orderly kind of progression is followed. None is now known in base 10, though I wouldn't be surprised if one is found. But I doubt that 196 will turn out this way either. It looks to me like it will probably continue to grow chaotically, and I don't think our mathematics is up to proving anything about it.