This kind of coercion is intended to work, but not implemented.
Arrays do not implement Deref, so the coercion &[T; n] -> &[T] is not a deref coercion and does not work in quite the same way as one. Instead, it's called an "unsized coercion" because it turns a sized type ([T; n]) into an unsized one ([T]).
That said, the language reference (which is not normative and may be outdated, but bear with me) lists the possible coercions, including the following (emphasis added):
T_1 to T_3 where T_1 coerces to T_2 and T_2 coerces to T_3 (transitive case)
Note that this is not fully supported yet.
&T or &mut T to &U if T implements Deref<Target = U>.
TyCtor(T) to TyCtor(U), where TyCtor(T) is one of
&T
&mut T
*const T
*mut T
Box<T>
and where U can be obtained from T by unsized coercion.
The last bullet, unsized coercion, is what allows &[T; n] to coerce to &[T]. Notably, this only describes one layer of referencing; it doesn't cover the &&[T; n] -> &[T] case (for which we also need Deref coercion).
Back to your non-working example:
let arr: &[i32] = &&&[1, 2, 3, 4, 5];
The intended coercion is &&&[i32; 5] -> &[i32]. We can work out how this coercion ought to work:
&[i32; 5] coerces to &[i32] by unsizing;
&&[i32; 5] coerces to &[i32; 5] by Deref;
- therefore,
&&[i32; 5] coerces to &[i32] by transitivity.
&&&[i32; 5] coerces to &&[i32; 5] by Deref;
- therefore,
&&&[i32; 5] coerces to &[i32] by transitivity.
But it doesn't. The quote above hints at why: under the transitive case, it says "Note that this is not fully supported yet". As far as I can tell, according to issue #18602, "not fully supported" is a hedge; it would be more accurate to say "unimplemented". So, for now, coercion via transitivity is not possible at all. Apparently this issue is not a high priority, probably because sized arrays aren't very common. (I suspect this might become a more common complaint when const generics land, since that may make arrays more useful.)
So why does arr.first() work? Well, the "auto-dereferencing rules" used to find methods invoked with the . (dot) operator are different from the coercion rules. Autoderef is similar to manually dereferencing any number of times until you get something (that can be coerced to a type) with the given method. This means you don't need transitive coercion to find method calls through autoderef.
Further reading
RFC #401 describes intended semantics of most coercions, but has never been fully implemented. Issue #18602 tracks the status of transitive coercion.
The Rustonomicon also has a chapter on coercions and appears to agree with the reference book.