There is likely no point in which it stops physically counting. It doesn't have to; the account resets monthly therefore unless you got 1m in a month while the digits may read a "grand total" which is just an addition problem the actual amount attributed to your account is far, far less.
You'd have to ask the programmers. Only they would know how it's done on the back end. I would think that they would either use a database query to sum up all the xper points on each request or they would tally it and store it in a database field.
I would think it would be the latter more likely because such a query would be computationally expensive and because it takes a while for xper to show up after you enter a question. If the value is stored in a database field it depends on the RDBMS they use and the SQL data type they specified (INT, BIGINT) and whether it was signed or not.
Just for shits and giggles, let's assume that they used the largest available SQL data type (BIGINT). On MySQL and SQL server (and I would think most other commercial quality relational databases) that would be an 8 byte unsigned integer which can hold a maximum value of 2^64. So hypothetically you could accumulate 18446744073709551616 xper before the counter would wrap around.