For the sake of the answer, we'll consider a hypothetical format where all cards since the beginning of the PTCG are usable. This is because Lv. X and Pokemon Break cards cannot be used in official formats, as none allow for the use of Diamond/Pearl and X/Y sets in correlation with one another. I also won't be including any images, but instead the Bulbapedia article for the following card.
In addition, a big thanks to Staka~ for answering my question here, as then it'd have taken me far longer to research if this is even possible at all. Alright, time to get on with it, shall we?
Pokemon Cards with the MOST Attacks
Long story short, it's 6 attacks
How? Well, by applying a Lv. X card to a traditional Pokemon card, then using a BREAK card on the same card, we are able to get a Pokemon card normally with 3 moves to 5 moves, at least in theory. There are two Pokemon I know of that this applies too:
- Raichu (Stormfront 8) <- Raichu Lv. X (Stormfront 99) <- Raichu BREAK (BREAKthrough 50)
- Machamp (Stormfront 20) <- Machamp Lv. X (Stormfront 98) <- Machamp BREAK (Evolutions 60)
The thing is, I'm not entirely sure if this is possible, as the Pokemon BREAK may only inherit the Lv. X card's attacks, rather than from both cards. As this can't be tested, due to Lv. X and BREAK cards being split across generations, and I have yet to find a source of some madman replicating this, I can't say which is true for sure. Regardless, we're going to move on as if both card pairing are able to use a combination of 5 moves total.
Also, you might've noticed I said the Pokemon Cards with the most attacks add up to 6 attacks. Well, that wasn't a typo, and that's all due to our good friend Ditto, more specifically, Ditto (XY Promo 40). With the use of its ability Metamorphosis Gene, Ditto is able to copy all attacks from an opponents Pokemon. Combined with its attack Stick On, Ditto can, in theory, inherit the moves of an opponents Lv. X BREAK Raichu/Machamp, and obtain 6 attacks.