First, let me explain what I mean by iterated triangle numbers:
The triangle of a number $n$ is the sum of all numbers from 1 to n, i. e. $1+2+...+ n$. Iterated triangle numbers are what you get if you repeat this process (that is, find the triangle of a number, take the result, and then triangle again).
Now, on to my question. If you repeat this process over and over again, do the last digits ever repeat? If you take 2 and find the triangle repeatedly, you get 3, 6, 21, 231, 26796, 359026206, 64449908476890321, .... The next few numbers in the series have the last digits ...186681, ...991221, and ...531031. The 100th iterated triangle of 2 also ends in ...453031 (yes I actually took the time to find that out).
The last digit is always either 1 or 6, but which one it is is quite erratic, and I know of no way to find the last digit of, say, 2 triangled 443 times, without finding the last digits of all numbers in the series up to it.