The shortest ECDSA signature I want to create ECDSA signature {r,s} where r is very low (for example 1) and can be encoded in DER-format in one byte.

How to calculate k value? Seems to me that it does not depend of anything. What is the value of k which produces r=1 ?

Note: I do not worry for the safety of my private key while publishing this signature.
What is the value of k which produces r=1 ?

It's impossible to know that. If you could derive r for arbitrary values then ECDSA would be fundamentally broken. The best you can do is grind k until you get an r that happens to have a short encoding.
For the sake of the exercise:
k: 55573144136627188774517374788342221967869962622835886499477787746883063622036
r: 771676860789419846973923839003663416737624455477806040640071960112246091

This nonce will be slightly smaller than most when encoded, but of course if you attempt to use it in a signature you have exposed your private key. If you generate this secretly you can only use the k value once and the result might be at best a couple of bytes difference in the encoded transaction.
It hardly seems worth it to be honest.