How many Secp256k1 signatures exist for a piece of data? ECDSA signatures are not deterministic. That is, random information is used in calculating the signature for a piece of data. Approximately how many unique signatures exist for a piece of data?

Another way this question might be phrased is: if I continually sign the same piece of data using random multiples of the generator point then how many signatures (on average) will I have to generate before finding a duplicate signature? 

And finally, if instead I start with a random multiple of the generator kG and then sign using (k+1)G, then (k+2)G, ... then will I ever produce the same signature?
Approximately how many unique signatures exist for a piece of data?


something like 2^256 (slightly less)


  will I have to generate before finding a duplicate signature?


It depends of the quality of you random generator. You must use "true" random generators, not pseudo-random.


  then will I ever produce the same signature?


Are you going to live eternally? The sun will go out by the time of when you repeat the signature. By the way: you can take value of 'k' as a function of your 'privateKey' and 'data-2-sign' 
See https://bitcointalk.org/index.php?topic=727918.0 for more info

By the way. The signature itself is 512 bits. R and S (256 bits each). Your private key can be recovered when you sign different data with the same 'k'. In this case 'R' value will be the same for both signatures. And 'S' value will differ.