THE PERFECT INFINITIVE is used with WOULD LIKE/LOVE/PREFER and one or two other verbs to refer to unreal past situations; things that never happened; a missed opportunity, etc. There are two distinct structures for this usage.
“Would you like to have been born with a special power, such as mind reading?”
“Would you have liked to be born with a special power, such as mind reading?”
WOULD LIKE TO HAVE DONE expresses a present regret, desire or intention about a past event, whereas WOULD HAVE LIKED TO DO expresses a past regret, desire or intention about a past event.
“I would like to have gone to the party.” (a present desire about a past event)
“I would have liked to go to the party.” (a past desire about a past event)
There’s a difference in principle. In practice, however, many speakers will use one or the other without making any such distinction.
“I would like to have gone/would have liked to go to the party.”
In the second case though, you could, from a logical point of view, also say ‘I would have liked to have done it’ as both the regret and the event are past and it more explicitly echoes the past tense element of ‘would have liked’.
“I would have liked to go/to have gone to the party.”
The so-called DOUBLE PERFECT (INFINITIVE) is commonly used in informal speech.
“When I was a child, I would have liked to have learned to play the guitar.” (but my parents wouldn’t let me do so)
The two forms are generally used interchangeably and the extra perfect infinitive does not change the meaning.
“I would have liked to see/have seen her at the party.” (but she didn’t come)
“I would have liked to know/have known your father.” (who is likely no longer available to be known)
Being often stigmatized as incorrect, the structure is still understandable, and admittedly common. Erroneous or not, such phrasings have certainly been in use for a long time.
“Peggy would have liked to have shown her turban and bird of paradise at the ball.” [Thackeray]
Garner’s Modern American Usage states that “would have liked should invariably be followed by a present-tense infinitive — hence would have liked to go, not would have liked to have gone.”
“He would have liked to have taken his phone with him, the meeting was way longer than he expected.”
That said, Garner acknowledges that the erroneous phrasing is very common. In fact, Google Books claims 386,000 written instances of ‘would have liked to have seen’, for example, enough to call it a stylistic choice.
“And I would have liked to have known you but I was just a kid.” [Candle in the Wind by Elton John/Bernie Taupin]
One conclusion is that it’s just a pattern which has emerged and become frequent enough so that it’s acceptable in everyday spoken English, if not in academic or professional contexts.
“It is doubtful whether the Labour members would have dared to have gone further.”
Bob Ladd states: “After past tenses of hope, fear, expect, and the like, the perfect infinitive is used, incorrectly indeed and unnecessarily, but so often and with so useful an implication that it may well be counted idiomatic.”
“Washington and its diplomatic partners would have liked to have seen even more from Beijing in this period.”
That implication is that the thing in question did not in fact come to pass, and the economy of conveying this without a separate sentence compensates for lack of logical precision.
“As a teenager, I would have loved to have known my father.” (who died before I was born)
Logic supports such a structure only if you have enough backward steps in your flashbacks, this past-referring-to-further-past sort of thing. However, quite often use of so-called double perfect remains questionable.
“Would you have liked to have been president from 1862-1864?” [The New York Times, 2008]