This phenomenon is better known as "elision".
In formal French, only a small set of words is subject to this rule. All are function words and most of them end in the unreduced vowel schwa /ə/ (the one exception is la).1
le la ; de ; je ; me te se ; ne ; que (and composite forms of que )
The edge cases are the fixed elisions c'est, s'il, s'ils, not relevant to the following discussion.
This enumeration leads to the observation that in your second example, there's only one possible site of elision: l'homme.
If you're expecting homm'appelle, let me clarify. Many of the e muets that you see at the end of words like homme are never pronounced in some dialects, including Parisian, whereas they are always pronounced in other dialects — except that before another vowel they are indeed elided.
So you would get /lɔmapɛl/. But these cases aren't written with an apostrophe (outside of old song lyrics, e.g. « encor' à » here). Grammar books may not treat this rule under the same heading.
But what happens if we do have two of these eliding words in a row?
Consider that the rule for eliding is that the reduced vowel must precede an unreduced vowel.2
Je aime ça → J'aime ça
But you'll notice that none of these eliding words begins with a vowel itself. That means that there will never be chaining between them such that the rule applies twice.
That is, if you have two in a row, only the second will be elided:
Tu parles de le éléphant → Tu parles de l'éléphant
Je me appelle Coco → Je m'appelle Coco
However, in spoken French, this is often spoken fast enough that you do elide both reduced vowels: d'l'éléphant, j'm'appelle. Reduced vowels tend to be dropped at high speeds in all languages!
Just keep in mind that writing it with two apostrophes is an informal stylistic choice.
1 In informal French, there are fewer restrictions on elision. For example, tu as → t'as even though /y/ is not a reduced vowel. To my knowledge, this still only happens with function words.
2 As aCOSwt noted, if the subject and verb and inverted, as in Suis-je arrivé ?, you won't write the apostrophe. The unreduced vowel will still drop in pronunciation.