Apparently not.

The series clearly ceases to increase as soon as any term contains a zero.

If we start with 1, the series goes **eleven** terms before hitting the term 102,

and that is long compared with other small starting numbers. For the first 100

starting numbers the series seems to stop growing at either 102 or 202:

1 2 4 8 16 22 26 38 62 74 **102**

For starting values less than 1000 the latest first occurrence of zero is the **28**^{th} term.

The lowest such starting value is 187. There are two others, namely 248 and 264,

and the zero-term in each case is 4506:

187 243 267 351 **366** 474 586 826 922 958

1318 1342 1366 1474 1586 1826 1922 1958 2318 2366

2582 2742 2854 3174 3258 3498 4362 **4506**

248 312 318 342 **366** 474 586 826 922 958

1318 1342 1366 1474 1586 1826 1922 1958 2318 2366

2582 2742 2854 3174 3258 3498 4362 **4506**

264 312 318 342 **366** 474 586 826 922 958

1318 1342 1366 1474 1586 1826 1922 1958 2318 2366

2582 2742 2854 3174 3258 3498 4362 **4506**

These show two different paths to the term 366, after which the term 4506 is inevitably reached.

Next in length is the starting value of 3237326, which has **29** terms:

3237326 3241862 3244166 3247622 3251654 3255254 3261254 3262694 3278246 3294374

3312518 3313238 3314534 3316694 3328358 3345638 3371558 3384158 3395678 3531758

3544358 3573158 3585758 3753758 3841958 3876518 3916838 3947942 **4002374**

Next is 3515987, which goes a whopping **33** terms:

3515987 3553787 3641987 3678275 3748835 3829475 3889955 4278755 4357155 4367655

4443255 4452855 4484855 4587255 4643255 4657655 4783655 4884455 4986855 5332455

5341455 5347455 5389455 5497455 5623455 5641455 5653455 5698455 5914455 5932455

5959455 6161955 **6170055**

No starting point less than 10 million has a longer sequence, although 5112691 also

reaches 33 terms before hitting a zero:

5112691 5113231 5113321 5113411 5113471 5113891 5114971 5116231 5116411 5116531

5116981 5119141 5119321 5119591 5121616 5121976 5125756 5136256 5141656 5145256

5151256 5152756 5163256 5168656 5211856 5214256 5216656 5227456 5244256 5253856

5289856 5462656 **5505856**

There may be a proof that a zero must be reached -- and the OP suggests there is one --

other than the intuitive observation that unless some mechanism actively prevents zeros,

and it's hard to imagine such a mechanism exists, eventually the terms have so many digits

that the probability that none of them is zero becomes vanishingly small. That is, for the series

to be unbounded, the terms will reach one million digits in length. For an endless number of

such terms not to contain a zero is unimaginable.