I have mentioned it elsewhere in the thread, so I feel slightly uncomfortable mentioning it repeatedly because I don't want be a walking advertisement, but it's Trupanion.
Definitely not an advertisement. For me with my dog there is no $25/mo plan. The lowest it goes is $60.95/mo, with a $1000 deductible.
Trupanion covers 90% after the deductible. A $3k bill would still require me to pay $1300 ($1700 savings), which I could have just saved myself by stocking some money away. $1700/$61=28 months. I've had my dog for more than 28 months and haven't had any $3k bills.
Of course, if my dog has a $20k bill, it makes more sense, but the chances of that are pretty low given how I know my dog. It becomes an emotional decision, which is my point and what they are counting on.