1496 is the smallest number whose trajectory under A053392 increases without limit.
More terms than usual are shown in order to display the onset of exponential growth.
Proof that this grows without limit, from Hans Havermann, Nov 01 2019: (Start)
One way to prove that the trajectory of a number under repeated application of the map defined in A053392 increases without limit is to show that there exists a term containing a nonfinal substring of three adjacent 9's.
If such a substring in term n is followed by a 0, it will grow to a nonfinal substring of three adjacent 9's followed by a 1 in term n+2: *9990* > *18189* > *99917*
If such a substring in term n is followed by a digit d that is neither 0 nor 9, it will grow to a nonfinal substring of four adjacent 9's in term n+2: *999d* > *18181(d1)* > *9999d*
If such a substring in term n is followed by another 9, then it is a substring of k 9's for k >= 4, which will grow to a substring of (2k3) 9's in term n+2: *[9]^d* > *[18]^(d1)* > *[9]^(2d3)*
Putting those together, such a substring in term n will grow to four adjacent 9's by term n+4; to five adjacent 9's by term n+6; to seven adjacent 9's by term n+8; ... to 2^k+3 adjacent 9's (see A062709) by term n+4+2k, regardless of what happens in the rest of the number.
In the present sequence a(41) contains two nonfinal substrings of three adjacent 9's. QED (End)
[Argument corrected and completed by David J. Seal, Nov 05 2019.]
