Simplest version of anapanasati Anapanasati is usually described as "mindfulness of breathing" but I've seen a member of this site describing it as "mindfulness with breathing".

Sati or mindfulness (as seen in the parable of the bowl filled to the brim with oil, of SN 47.20) is keeping the mind on the task, and not allowing the mind to become distracted.

What is the simplest version of anapanasati?

I found a very simple instruction set in SN 54.3 (quoted below). There may be other suttas too. But I find this description from SN 54.3 insufficient and requires elaboration.

While breathing, what is it that one is mindful of? Is one mindful of the breath, or mindful of something else?

The sutta states: "They practice like this: ‘I’ll breathe in observing letting go.’ They practice like this: ‘I’ll breathe out observing letting go.’"

What does "breathe in observing letting go" and "breathe out observing letting go" mean? Letting go of what? Do you let go, or do you observe the letting go? How do you observe the letting go (rather than letting go)?

What is the simplest instruction for the practice of anapanasati? Especially for people who are not well-versed in the teachings and maybe even non-Buddhist? Such people may not have the patience for lengthy and complicated instructions.

From SN 54.3:


  “Mendicants, when mindfulness of breathing is developed and cultivated
  it’s very fruitful and beneficial. And how is mindfulness of breathing
  developed and cultivated to be very fruitful and beneficial?
  
  It’s when a mendicant has gone to a wilderness, or to the root of a
  tree, or to an empty hut. They sit down cross-legged, with their body
  straight, and establish mindfulness right there.
  
  Just mindful, they breathe in. Mindful, they breathe out. …
  
  They practice like this: ‘I’ll breathe in observing letting go.’ They
  practice like this: ‘I’ll breathe out observing letting go.’
  
  Mindfulness of breathing, when developed and cultivated in this way,
  is very fruitful and beneficial.”
The SN 54.3 linked is not a "simple version". The link is merely abbreviated because the 16 stages are listed in previous suttas.

If per SN 47.20, mindfulness is keeping the mind on the task, since the body breathes automatically via its own nature, how can breathing be a ‘task’ to keep the mind on? There can be no such thing as mindfulness ‘of’ breathing. 

As for Sujato’s translation of “letting go”, this is the 16th step. Letting go of the 16th step is a result of 13th step experiencing impermanence. The simultaneous knowing of breathing occurs in the background.

The Pali here is paṭinissaggā. When impermanence &amp; not-self is clearly experienced at 13th step, the 16th step is paṭinissaggā, which means to give up viewing anything as self. Ajahn Buddhadasa translates it as throwing or tossing back. 

Paṭinissaggā above is the final step of practice. However, the type of letting go that serves as the initial step of practice is called vossagga. MN 118 at the end says each factor of enlightenment matures as vossagga. SN 48.9 &amp; 10 say jhana is reached by making  vossagga the meditation object.

Vossagga is letting go of craving &amp; ambition. This is the supramundane meditation object or task of mindfulness.

If you try to focus on the breathing, the progress will be very limited. The focus is on vossagga. The task is vossagga. Mindfulness is mindful of vossagga.

When there is mindfulness of vossagga, the knowing of the breathing will occur automatically, without an act of will (AN 11.2). This is why anapanasati means mindfulness with breathing.