This paper presents some results concerning the search for initial values to the so-called
3
x
+
1
3x+1
problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of
n
n
, the maximum value of the function iterates is bounded from above by
n
2
f
(
n
)
n^2 f(n)
, with
f
(
n
)
f(n)
either a constant or a very slowly increasing function of
n
n
. As a by-product of this (exhaustive) search, which was performed up to
n
=
3
⋅
2
53
≈
2.702
⋅
10
16
n=3 \cdot 2^{53}\approx 2.702 \cdot 10^{16}
, the
3
x
+
1
3x+1
conjecture was verified up to that same number.