All good points. But how many reCaptchas occur per day? Google could also use page hits to google as well. But this is parallel to my idea to use a separate captcha to mine BTC. Let's call it caBTCha.
Likely not enough. Lets say you gained 100% of the market and sustained a load of 100 million captchas per second. Just to put that into perspective that is an average of almost 3000 captas per internet user per day.
If you could solve a double hash using only 50 math captchas (which I think is unrealistic) per double hash and with no error (100% user accuracy) you would make almost 2 cents per day. That is before your cost of running a server network capable of generating 3000 captchas per person on the internet daily.
The reality is that a single hash is worth almost nothing.
At current difficulty it takes (2^32)*1.3 million = 5.58E+15 hashes to form a block. That puts the value (with current price & BTC reward) at 4.92526E-14 per hash.
Another way to look at it is that it takes 203,034,817,629 (203 trillion) hashes to generate 1 penny of value.