A recent discussion about the evolution/devolution of languages often left me wondering this: When someone says (speaking relatively, of course) that "This language is complex" or "This language is not very complex," what could we possibly be talking about other than grammar?
The possibilities I can think of are: diction, vocabulary, orthography, alphabet/writing system, and pronunciation.
Is, for example, Spanish less complex than Italian based strictly on having a more limited vocabulary? Or is Mandarin Chinese more complex than Italian based strictly on the relative complexity of its graphemes? Does "grammar" trump all when it comes to making such summary comparisons?
My initial answer to this question was "yes, of course it does," but now I am uncertain.

What are the leading examples of languages that illustrate this point?
: in which language is the language described? In it itself or in some other language?
What if we distinguish between “complexity” and “difficulty”? I tend to see the incoherence of spelling as, at first, difficult, and then potentially complex only in so far as the difficulty is systematically intricate. In other words, pronunciation and spelling are the leading perplexities of certain languages, but complexity is still mostly about the grammar. What are the leading examples for languages where spelling (and/or pronunciation) is a major complicating factor? Irish?
@Donald Remero: I don’t know if it is leading, but Danish is one example. There are so many exceptions that it must be learned by heart. The grammar is simpler than English; for instance there is no difference between first, second, third person, singular/plural for present, in contrast to English and German. Example: http://en.wiktionary.org/wiki/er#Danish
Remero: I don’t know if it is leading, but Danish is one example. There are so many exceptions that it must be learned by heart. The grammar, on the other hand, is simpler than English; for instance there is no difference between first, second, third person, singular/plural for the present tense, in contrast to English and German. Example: en.wiktionary.org/wiki/er#Danish. But it all hinge on the definition of “complex”.
@Donald Remero: I don’t know if it is leading, but Danish is one example. There are so many exceptions that it must be learned by heart. The grammar, on the other hand, is simpler than English; for instance there is no difference between first, second, third person, singular/plural for the present tense, in contrast to English and German. Example: http://en.wiktionary.org/wiki/er#Danish. But it all hinge on the definition of “complex”.
@Peter: for me the grammar complexity is the number of characters (or book pages needed) to describe language http://en.wikipedia.org/wiki/Kolmogorovcomplexity. Here arises the question – in which language it should be described http://en.wikipedia.org/wiki/Backus%E2%80%93NaurForm
: I am not sure I understand what you mean by grammar complexity. Is the grammar complexity for a particular text, say the book “Mein Kampf” or the book “Das Kapital”. Or is it for some text (normal prose) that describes, say, the German language?
: I am not sure I understand what you mean by grammar complexity. Is the grammar complexity for a particular text, say the book “Mein Kampf” or the book “Das Kapital”. Or is it for some text (normal prose) that describes, say, the German language? As far as I understand the Kolmogorov complexity is for a particular text.
Interesting discussion. Going to ponder on this for a bit. It seems there is a tendency among these mainly computer-based analytical tools to discuss “complexity” largely in terms of how many steps, rules, or possibilities there are in the generation of a valid statement. Perhaps “difficulty” is more properly applied to the mechanics of spelling, pronunciation, and the like. I am not too interested, though in belaboring the mere definition of ‘complex.’ Though, perhaps difficulty/complexity is a useful distinction.
I am sure that linguists have already definitions for each of kind complexity as described in OP. For example computer scientists think that Chomsky is a computer scientist because of his contributions to theory of grammars/complexity although Wikipedia says he is a linguist http://en.wikipedia.org/wiki/Noam_Chomsky. So I think he has already worked in this field. I should read his books.