@gonutz i decided to build on your idea of using a computer to do the work for us!
here's the result, with a base of 26. (my finger hurts) heartbeat2.png i haven't proved for sure that my strategy creates the highest number, but it's pretty darn high.
one thing to notice is that you can go into an endless loop of applying the third card. if the remainder of the number divided by 3 is 2, then every time you apply the card it'll result in another number with a remainder of 2. or, to put it mathematically, ``` if n % 3 = 2, then (n * 2 + 1) % 3 = 2. ``` this means that the number will never be divisible by 3!
of course, this only ever increases the number by a factor of 2. the second card can increase the number by so much more, but it has the potential of throwing us off of our remainder of 2 journey. that's why i wrote a quick program that checks if the second card throws us off track, and applies the third card otherwise. and so we end up with this number!
i did add a part to my program checking if the fourth card was useful, but it never seemed to be. the first card wasn't that useful either, since it would often lead to dead ends where the only way forward is to drop down using the second or third cards. (there are very few cases where decrementing the counter is beneficial to us, but note that they do exist, i think). i do think there's a possibility these cards are useful, but it's kinda low and it would take some more work to figure out.
edit: noticing quite a few possibilities with fourth card -> second card. probably the earliest one is by starting off with 32, fourth card (128) then second card (821), and continuing using my above strategy, resulting in a number that is 11 digits long! there are also a million other later ones that use fourth to second, like 392 -> 8651 or 4712 -> 84881; tons of room to go higher than my original result.