OP asked about shuffling multiple times and having the deck become less random intentionally. AFAIK, the only way to do this deliberately is with the riffle, and it only works from a sorted deck.
A deck can indeed end up more ordered after one or more shuffles than it was before. But if the shuffle has any randomness to it, this is coincidence, and it will only happen as often as it is likely, which is not much. The same way you do have a probability of flipping 10 heads in a row, you can accidentally organize a deck. But the chance of flipping 10 heads in a row is (you know already) slightly lower than one in a thousand. The chance of just happening to organize a randomized deck after one or multiple shuffles is "a lot" lower than that.
A deck can indeed end up more ordered after one or more shuffles than it was before. But if the shuffle has any randomness to it, this is coincidence, and it will only happen as often as it is likely, which is not much. The same way you do have a probability of flipping 10 heads in a row, you can accidentally organize a deck. But the chance of flipping 10 heads in a row is (you know already) slightly lower than one in a thousand. The chance of just happening to organize a randomized deck after one or multiple shuffles is "a lot" lower than that.