  The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The
total stopping time \sigma_\infty (n) for a positive integer n is the number of
iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1
is never reached. The 3x+1 conjecture states that this function is finite. We
show that infinitely many n have a finite total stopping time with
\sigma_\infty(n) > 6.14316 log n. The proof uses a very large computation. It
is believed that almost all positive integers have \sigma_\infty (n) > 6.95212
\log n. The method of the paper should extend to prove infinitely many integers
have this property, but it would require a much larger computation.
