Do there exist any numbers such that any in-order combination of digits is prime (including the original number)? I was inspired by this question, in particular, where removing any digit of a prime yielded another prime. My question is, are there any numbers that this holds true for, and will continue holding true for if digits continue to be removed?

I have a strong feeling that this is false. However, I'm very interested in seeing a proof for the falsehood of this, or maybe even a proof that one does exist but is not computable within reason.
If you're asking what I think you're asking, it shouldn't be hard to work through it. First, since you eventually get down to one digit, only prime numbers can be digits; 2, 3, 5, and 7. But 2 and 5 are no good, since any 2-digit number ending in 2 or 5 will be composite, so we just have 3 and 7. But we can't use either digit twice, since if we remove all the other digits we'll get 33 or 77, both composite. So Matthew's 37 (and 73) is the only answer of more than one digit. 

If you choose to make believe that 1 is a prime, then you also get 13 (and 31) and 17 (and 71), and you have to check whether the numbers 137, 173, 317, 371, 713, and 731 are all prime.
Found this when I went looking for numbers where any combination of their digits result in a prime, since earlier I happened to stumble upon this property of 3 digit numbers comprised of two 1's and one 3, those being 113, 131, and 311, which are all prime, and all satisfy the &quot;take any 1 digit and a prime will remain&quot; condition, though they all start with a crutch since they're comprised of only 2 unique digits. I also happened to find something interesting while testing the combinations of the digits in the number provided in the other post you linked, there are two of the six combinations which were not prime, 791 and 917, both of which had a 7 and a 3 digit number comprised of two 1's and one 3 as factors, 791 = 7 * 113 and 917 = 7 * 131, it's oddly coincidental how the example managed to connect to my number at all, but I thought it was really interesting. Also I'm 9 years late lmao