First, I want to point out (as user1873 did in the comments) that that map is not an accurate representation of the historical geography of the conflict.
- In image one, the green land was not Palestinian, but Ottoman and then British. In the 19th century, Palestine was a part of the Ottoman empire, but after Britain defeated the Ottomans in World War I, it became part of Britain's Mandate of Palestine.
- Palestinians rejected an offer to draw the borders according to the second map. After Britain withdrew from Mandatory Palestine, the UN proposed that as a partition plan. The Jews accepted that plan, but the Palestinians rejected it.
- In the third map, the green areas belonged to Jordan and Egypt. After Israel declared independence, all surrounding Arab nations invaded Israel at once. Israel won, signing armistice agreements with its neighbors. These agreements defined temporary borders, giving the West Bank to Jordan and Gaza to Egypt.
- The fourth map is almost accurate, but a little out of date. In 1993, Israel granted Palestinians sovereignty over land for the first time in the Oslo Accords. These agreements defined areas of Palestinian sovereignty in much of the West Bank and Gaza. But in 2005, Israel withdrew from Gaza entirely. The final map is slightly inaccurate because it still has Israel occupying part of the Gaza strip. However, that is no longer the case.
In short, the image should not plot any areas as green until the final map, as it wasn't until 1993 that the Palestinians held sovereignty over any land.
The fact that the map is inaccurate suggests that your prediction is incorrect. Israel is not going to take over all Palestinian land within a few years. In fact, the trend has been moving in the opposite direction. Israel's peace offers to the Palestinians have become progressively more generous, giving the Palestinians more land. Over time, the Palestinians have been offered access to more land in final settlement offers, not less.
You also asked why Israel would or would not expand into other countries. There are a few reasons for this. First, it does not want to. Israel has enough demographic problems controlling Palestine, which is why Israel has consistently offered to give Palestinians land in exchange for peace. West of the Jordan River, around half of the population is Jewish and half is Arab. If Israel kept those areas, it would have to choose between being a Jewish and a democratic country. Capturing other Arab areas would only exacerbate Israel's demographic problems.
Furthermore, Israel has peace agreements with Egypt and Jordan. These peace agreements defined the borders between the countries and ended the conflict between them. Israel also borders Lebanon and Syria. Though Israel does not have peace agreements with these countries, it has no motivation to capture their territory. In fact, Israel withrew from southern Lebanon in 2000, which it had occupied as an attempt to combat terrorism coming from Hezbollah. Israel had also offered the Golan Heights (which it captured from Syria in the Six Day War) back to Syria several times in exchange for peace.
Your question is premised on the assumption that Israel's goal is to capture more land, but its history and peace offers indicate the opposite. Rather than capturing land, Israel has a policy of giving up land in exchange for peace agreements from its neighbors.