Your probability of getting this exact sequence would be (1/6)^6. This is a geometric series with p = (1/6)^6.
The expected value for the number of attempts you'd have to make would be, therefore 6^6 = 46,656. But This would assume discrete attempts: you'd roll 6 dice, check to see if you got the desired sequence, if not, roll 6 dice again. So your expected value would be 46,656 attempts * (6 rolls per attempt) = 279,936
But this is not your exact situation. In your case, we have to roll until we get a 1, then get the sequence (2,3,4,5,6) afterwards. And if this was a fail, we just look for the next 1. So I'm thinking I'm overthinking it. The answer is just 6^6 = 46,656.
