Q: What are the implications of the recursive joining of Promises in terms of Monads? I know that Javascript's promises are technically neither functors nor monads in the sense of Haskell, because (among other things)


*

*they include a bind operation that falls back to map when a pure function is passed in (and thus has an ambiguous type)

*both the Promise constructor and resolve (aka return) join nested promises recursively


The first issue can easily be bypassed by always providing a function with the right type a -> Promise b.
The second issue obviously violates the parametricity trait of parametric polymorphic functions, i.e. one cannot construct a m (m a) structure. But what would this structure mean in the context of promises/asynchronous computations? I cannot think of a meaningful semantics for Promise (Promise a), where Promise is a monad. So what do we lose? What are the implications of the recursive joining?
Provided we are pretty pragmatic (and that's what we should be when we're programming Javascript), can't we claim that a Promise is a monad in Javascript if we take care of the edge cases?

A: 
The first issue can easily be bypassed by always providing a function with the right type a -> Promise a.

Or by not using then as the bind operation of the monad, but some type-correct ones. Creed is a functionally minded promise library that provides map and chain methods which implements the Fantasy-land spec for algebraic types.
The second issue can be bypassed as well with the same approach, by not using resolve but fulfill instead, and the static of method as the unit function.

But what would this structure mean in the context of promises/asynchronous computations?

It's a promise for a promise for a value. Not every constructible type needs to be "meaningful" or "useful" :-)
However, a good example of a similar type is provided by the Fetch API: it returns a promise that resolves to a Response object, which again "contains" a promise that resolves to the body of the response.
So a Promise (Promise a) might have only one success result value, which could as well be accessed through a Promise a, however the two levels of promises

*

*might fulfill at different times, adding a "middle step"

*might reject with different causes - e.g. the outer one representing a network problem while the inner one represents a parsing problem

Notice that the Promise type should have a second type variable for the rejection reason, similar to an Either. A two-level Promise err1 (Promise err2 a) is quite different from a Promise err a.

I know that Javascript's promises are technically neither functors nor monads in the sense of Haskell

You haven't mentioned the biggest issue yet, however: they're mutable. The transition from pending to settled state is a side effect that destroys referential transparency if we consider execution order, and of course our usual use cases for promises involve lots of IO that isn't modelled by the promise type at all.
Promise.delay(50).then(() => Promise.delay(50))
// does something different than
const a = Promise.delay(50); a.then(() => a)

Applying the monad laws is fun and occasionally useful, but we need lots of pragmatism indeed.

A: 
I know that Javascript's promises are technically neither functors nor monads in the sense of Haskell

Not only in the sense of Haskell, in any other way as well.

  
*
  
*they include a bind operation that falls back to map when a pure function is passed in (and thus has an ambiguous type)
  

there is no bind operator provided by JS native promises

  
*
  
*both the Promise constructor and resolve (aka return) join nested promises recursively
  

I presume you mean unwrapping "theneables", i.e. calling functions stored under then prop whenever there is such function.

The first issue can easily be bypassed by always providing a function with the right type a -> Promise b.

This would not resemble map e.g. when map(f) is used for 
f = x => {then: a => a}.

The second issue obviously violates the parametricity trait of parametric polymorphic functions, i.e. one cannot construct a m (m a) structure. 

Indeed.

But what would this structure mean in the context of promises/asynchronous computations? I cannot think of a meaningful semantics for Promise (Promise a), where Promise is a monad. So what do we lose? What are the implications of the recursive joining?

You need to allow storing arbitrary values. Promises are not allowed to store theneables (without unwrapping), which is the problem. So you need to change semantics of both objects and methods. Allow objects to store theneables without change and implement .bind aka .chain that unwraps (or joins) theneables precisely once - no recursion.
This is what creed does for promise-like objects and cpsfy for callback-based (aka continuation passing style) functions.


Provided we are pretty pragmatic (and that's what we should be when we're programming Javascript), can't we claim that a Promise is a monad in Javascript if we take care of the edge cases?

Writing safe, succinct and composable code is pragmatic. Risking introduce subtle bugs via leaky abstractions that might crash critical software with far going consequences is not. 
Every edge case is a potential source of such risk.
In that respect, claiming that Promise is a monad does more harm than help, beside being incorrect. It does harm because you cannot safely apply monadic transformations to promises. E.g. it is unsafe to use any code conforming to the monadic interface with promises as if they were monads. If used correctly, monads are there to help abstracting and reusing your code, not to introduce lines of checking and hunting for edge cases.
