Let the Games Begin: Bibi has his work cut out for him
He might try to form a coalition of right-wing and religious parties; a government backed by his party, the pro-settlement Jewish Home party (12 seats) and the two Hareidi (fervently Orthodox) political parties, the Sephardic party Shas (11 seats) and United Torah Judaism (a party advancing the interests and point of view of the Ashkenasi yeshiva and Chasidic sectors, with 7 seats) would control 61 votes, the minimum majority. But that coalition would be extremely unstable. For example, there is no love lost, between Shas and the Jewish Home--the rabbinic leader of Shas, Rabbi Ovadia Yosef, described Jewish Home as anti-Torah, even though its voter base is made up largely of Orthodox Jews. That is because Jewish Home leader Naftali Bennett favors curtailing the exemption of Hareidi yeshiva students from the draft.
Moreover, Jewish Home (joined by many members of Likud-Beiteinu) would try to obstruct any movement toward renewed negotiations with the Palestinians that might lead to a Palestinian state, which would pit the government against the rest of the Israeli political spectrum, a majority of the Israeli public and the United States.
Prime Minister Netanyahu might also try to form a broad-spectrum government, including Yesh Atid (19 seats), the new centrist party headed by broadcast journalist Yair Lapid, Labor (15), Kadima (2) and the Tnua party of Tzipi Livni (2). However, Labor has loudly declared its unwillingness to sit in a Netanyahu-led government. That means such a coalition would require another substantial party to reach a majority. The anti-Zionist Communist and Arab parties (11 total seats) are out. Yesh Atid campaigned on a platform of ending yeshiva draft exemptions and subsidies, so that eliminates Shas and United Torah Judaism as well. That just leaves Jewish Home, which as noted above strongly opposes the willingness of Yesh Atid to renew negotiations with the Palestinians.
So there is no clear path to a stable Israeli government based on these election results. Even if Bibi manages to cobble something together, one should expect new elections sooner rather than later.