Were the Jews expecting God Incarnate?

No.

I hang out asking questions to Jews in their Jewish stack over flow. Neo Judaism is very opposed to the idea of God incarnate or take any human form. To them, God is spirit.

https://judaism.stackexchange.com/questions/10349/if-god-never-incarnate-how-come-he-sat-in-front-of-abrahams-tent-and-appeared

https://judaism.stackexchange.com/questions/9187/does-g-d-get-incarnated

To Jews, God the almighty is not human and CANNOT take human form, which is strange given that he's almighty and all. I guess He can do anything that doesn't include taking human form.

I got tons of downvotes there for even asking who Jacob wrestled with, or who did Abraham see under the oak of Mamre.

So the answer is no. Jews do not expect God to ever be incarnate and if you don't believe me, go ahead ask the same question in Jewish stack overflow.

In fact, why ask questions on what Jews believe here? It's not fair to them I think. Why not ask them themselves on their forum?

NB: They seems to be pretty pissed that I even asked that in their forum. I do not know whether I am wrong because I don't know hebrew or because they too, like us, have their own religious filter they don't want to change. My impression is, I don't know hebrew, I won't understand that anyway, and well Torah is not for me anyway because I am not jewish so don't worry about it. Maybe I just don't get along with people well.

No they were not.

"The messiah" is a quite marginal concept in the Hebrew Bible. It's actually a workaround, a relatively new concept that started to emerge when things have gone bad (i.e. First Temple was destroyed). It's so marginal that the specific term "the messiah" does not appear even once in the Hebrew Bible (as opposed to the term "a messiah" which occurs in the Hebrew Bible 39 times, and is usually translated by KJV as "an anointed (one)", except for two occurrences in Daniel (Daniel 9: 25-26) that KJV chose to translate as "the messiah" - even though the Hebrew text simply says "an anointed one", not "the anointed one". Some modern translations have corrected this mistranslation.)

The above might sound quite annoying to the average Christian, and would surely entail a fair amount of downvoting. But bear in mind that Jews (and Christian fundamentalists alike) today, often use a similar argument in order to explain to Muslims why their claim on Jerusalem is bogus - why it's political more than religious: Jerusalem is mentioned zero times in the Koran, while it's mentioned 669 times in the Hebrew Bible. And while the Hebrew Bible never uses the term "the messiah", it's Greek equivalent - "Christ" - is used in the New Testament 240 times. The pattern is quite clear: if a matter is important to you, you mention it over and over. If it isn't important to you, you rarely mention it, or ignore it altogether (and of course, by your very silence on that matter, you leave to the future abusers of your writings the possibility for finding "hidden prophecies" or "hints" in your every second sentence).

The actual term "the messiah" started being used among Jews only after the latest book in the Hebrew Bible was written.

So, the concept is in the Hebrew Bible, it emerged after the destruction of the First Temple, and it's rather marginal in the Hebrew Bible.

Why? When the Jews lost their temple and their sovereignty over their land, some prophets came up with prophecies that their temple and sovereignty will be restored. Be restored to what state? What was the idyllic period in the history of the Israelites? Of course: David's days. This is why the prophets were talking about a King that will continue David's legacy, even by being his descendent. That King of course would be a human being. It's never stressed directly in the Hebrew prophecies, because it was so obvious - the prophets never imagined that there would come along in the future a religion that would try to hijack their words - they were surrounded by pagan nations that showed no interest whatsoever in the Israeli faith - but even though it's not stated directly in the prophets' words, it's still very clear from an honest reading of the true messianic paragraphs in the prophets' writings, that the king they are envisioning will be nothing but a human being - a very successful one though.

The Christian evolution of the term "the messiah":

Gentile Christians took it to a totally different place. While the first Jewish followers of Jesus (while he was still alive) sincerely believed that he would be that earthly king that would drive away the Romans, which is why they referred to him as "the messiah" in its traditional Jewish sense - Gentile Christians first used the figure of Jesus as an agent of sin cleansing. This was mainly the result of the theology preached to them by Paul. It made a lot of sense to the Gentiles - since they had no background in what the Hebrew Bible is about. To be a sin cleanser, Jesus of course had to be divine in some sense, which gradually developed to him being regarded as God (with the theology of trinity coming gradually along in the 3rd century as an effort to solve to tension between two Gods: Jesus and the Jewish God, and the concept of one God).

The Gentile Christians were not actually looking for "the messiah". The term makes no sense to them: it's a term that is all about restoring the national sovereignty of the Jewish people - but they are not Jews, so how could be Jesus sold to them as the King that restores the Jewish sovereignty over the land of Israel? That would be like selling TV's in the desert - there's no need for them there. So the Gentiles first got Jesus as God. Then in retrospect, they looked back at the origins of Jesus' story and tried to prove to themselves that the story is valid. What did they still see there, even through the oral stories that became more and more gentile and anti-Jewish? They saw a man who was admired by his followers as "the messiah" (not as God!). From that, the Gentiles concluded, "the messiah" is an honorary title, with no actual practical meaning. But to the Gentiles, Jesus is already God, hence he deserves to be given every honorary title. So let's call him also "the messiah", said the Gentiles - whatever it might actually mean.

Conclusion: The Jews never expected a divine messiah. It can be easily seen in the Hebrew prophecies found in the Hebrew Bible. The Gentiles first got Jesus as God incarnate. They also called him "the messiah" as an honorary title, imitating the Jews who called "the messiah" anyone who was believed to be able to restore their sovereignty, without actually understanding what the term meant to the Jews. To Christians, it's not that the messiah was understood to be God. It's God who got the title "the messiah".

Here's my brief survery:
1) The Essenes were expecting two human messiahs (see my comment under Mason Wheeler's answer)
2) OT sources like Daniel seem to expect some sort of divine figure to come and destroy Israel's enemies and restore Israel (look for the Daniel "Son of Man" passages). Look at the passage where Jesus tells his disciples for the first time that he is the messiah (Lk 9:20-26). He tells them he's the messiah but immediately explains that means he's to suffer rather than destroy Israel's enemies. Elsewhere in one of the gospels, someone asks Jesus if its time to destroy Israel's enemies or rise to arms (I don't remember where...or maybe I'm imagining it...) and Jesus rebukes them.
3) A second strand of biblical sources expect Elijah to come again. We see the roots of this in that Elijiah didn't "die" but was taken up to heaven in a chariot (Somewhere in 2 Kings?). We see this expectation in Mt 11:14, Mt 17:10, Jn 1:21--repeatedly Jesus is taken to be Elijah come again.

The Jews were a bit conflicted about what sort of messiah was to come, but they most likely did not expect Jesus having the certain Christology that we now know about.

Not everything was understood by everyone. We have the perspective of hindsight, but they did not have such a luxury.

In Matt. 22:41-46 (cp. Mark 12:35-37; Luke 20:41-44), the Lord Jesus Christ asked the Pharisees,

Saying, "What do you think of the Messiah? Whose son is he? They say to him, "The son of David." He says to them, "Then, how does David in the Spirit call him, 'lord,' saying, 'Yahveh said to my lord, 'Sit on My right hand until I make your enemies your footstool.''? So, if David calls him 'lord,' how is he his son?"

And no man was able to answer him a word, nor did anyone from that day forward ask him any more questions.

According to that conversation, the Pharisees could not understand how the Messiah is David's lord, which is of course because the Messiah is God.

Yes and No

There were some spiritually aware individuals among the Jews who were expecting the Messiah: Simeon and Anna. However, most Jews were almost certainly anticipating what Christians call the second coming: what is mentioned in Isaiah 2:1-4,

The word that Isaiah the son of Amoz saw concerning Judah and Jerusalem. And it shall come to pass in the last days, that the mountain of the LORD's house shall be established in the top of the mountains, and shall be exalted above the hills; and all nations shall flow unto it. And many people shall go and say, Come ye, and let us go up to the mountain of the LORD, to the house of the God of Jacob; and he will teach us of his ways, and we will walk in his paths: for out of Zion shall go forth the law, and the word of the LORD from Jerusalem. And he shall judge among the nations, and shall rebuke many people: and they shall beat their swords into plowshares, and their spears into pruninghooks: nation shall not lift up sword against nation, neither shall they learn war any more.

Yes they were, in fact the problem was not that they were expecting something less than God in the flesh, the problem was they were not expecting a lowly appearance. They expected his return in glory and judgment when he establishes peace forever -- what we now understand will happen at his second coming. It was his humanity, not his divinity, that tripped them up.