Some of you are looking too narrowly at what an ESA is.
It is not by definition non-task trained. It simply isn't required to be task trained in order for an animal to be an ESA.
Brain surgeons aren't required to know how to play golf, but they can if they want to. An ESA isn't required to perform tasks, but it can if you teach it to do so.
Here's an example:
I am disabled. I have an ESA cat. The cat is trained to retrieve small items from under furniture, especially hair rubber bands. My disability does not prevent me from getting the hair rubber bands myself. It was simply amusing and useful to teach the cat to do it for me. Retrieving hair rubber bands is a trained task, but because it does not mitigate my disability it would not justify calling my cat a service cat. Because it was also not possible to proof this cat to the point of reliability, he still wouldn't be a service cat even if I needed that particular task.
There are also home service dogs, who have the task training, but not the public access training. Many hearing dogs fall into this category. Legally they are still service animals. They simply are not used in public. But task training in and of itself isn't going to turn all ESAs magically into service dogs.
I just don't want to discourage anyone from training a dog to do anything at all (so long as it isn't harmful to the dog) because of semantics. Training can produce some helpful results, training is in itself fun (for both human and dog), and training is a very important source of mental stimulation and fulfillment for the dog.
So please, let's not tell someone they can't task train their ESA. Because they can! Maybe the training will change the dog's status, maybe it won't. That part doesn't really matter so long as the needs of both the dog and the person are being met.
Heck I started teaching Ruby some hearing dog tasks just for the heck of it and accidentally discovered an actual task related to my disability. I hear ringtones off and on all day so often don't recognize when my cell is actually ringing unless I've got it in my pocket. Keeping it in my pocket has been a workable solution. Unfortunately, I do not sleep with tight fitting clothes with pockets that will press the phone against me such that I actually feel it vibrate. I might have seriously missed out on something incredibly useful for managing my disability if I had believed I couldn't just train a task for the heck of it rather than to change a dog's status.
I train to make the dog pleasant company or helpful, to fulfill his/her needs for mental stimulation, and because it is fun. Not to change my dog's status. It's still the same dog, regardless of what label you slap on it. In a proper world it is still loved as much regardless of what label it wears ("nut" or "service dog" Ruby is still helpful and sweet).
Okay, so now I'm ranting. I just think sometimes there is a strong tendency to get too hung up in labels instead of the important stuff. Like task shopping. "I'm only interested in teaching my dog med reminders so I can call it a service dog, not because I actually need med reminders or because I'm just looking for something interesting to engage my dog."
I should step away. My dogs are becoming concerned. Didn't know this topic would excite me this much.