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:
$$4,5,6,12,53,222,231,416.$$
I'm currently checking between 2000 and 3000. I'll try to write something in a bit to check the various combinations of $10^n+3*10^i+3*10^k+3$.
Edit: 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 i previously mentioned and indeed there are lots, here a few:
100000030000000000000030000000000000000000000003 100000000000000000000300000000000000000000003003 100000000000000000000300000000000030000000000003 100000000000000000300000000000000000000003000003 100000000000000000300000000000000000000030000003 100000000000000003000000000000000000003000000003 100000000000000030000000000000000000300000000003 100000000000000030000003000000000000000000000003 100000030000000300000000000000000000000000000003 100000000000006000000000000000000000000000000003 100000000003030000000000000000000000000000000003 100000000000300000000000000000000000000030000003 100000000000300000000000000000000030000000000003 100000000003000000000000000000000000000000003003 100000000003030000000000000000000000000000000003 100000000303000000000000000000000000000000000003 100000000030000000000030000000000000000000000003 100000000300000000000000000000000000030000000003 100000000303000000000000000000000000000000000003
In particular I was always able to find a prime number of that form for n in the range [0,231]. I accidentally stopped my search at this range. I could probably run something over night, but it seems safe to say that even if there aren't an infinite number, there are a lot of them.