The problem seems not well specified but I noticed that you quoted an example of "10080 that slims to 7 after 7 steps" and you "want to know the largest number" (without any sensible constraint). By inference I think you meant to apply the constraint of "slims down to a single digit N after N steps"
The largest number that can be slimmed down to a single digit after that number of steps must surely result in 9 as it's the highest single digit (assuming we're still in base 10 here) so to find the number that slims to it, we fatten 9 up 9 times:
9, 18, 36, 72, 216, 648, 2592, 12960, 64800, 388800
9, 18, 36, 108, 324, 1296, 5184, 25920, 155520, 1088640
Thanks to @Jaap for the correction/pointing out a flaw in the algorithm that I didn't opportunistically take every possible occasion where a number could alter by an order of magnitude, for example
9 can only realistically go to 18
18 can only go to 36
but 36 could go to 72 if doubled or 108 if tripled, so we take the triple...
Of course if
we aren't in base 10, then we kinda need to decide what base we are in before we can go further - which gets back to the "you didn't specify a realistic upper bound of something" in your question..