As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:
$$4,5,6,12,53,222,231,416.$$
Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033 10000000000000000000000000000000003000000033 10000000000000000000000000000030000000000033 10000000000000000000000030000000000000000033 10000000003000000000000000000000000000000033
There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.
Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.