Where should one seek adventure in Jerusalem?

Definitely visit the old city, mount of olives, and visit the western wall during shabbat and watch the singing and dancing happening then. Also, visit the amazing Austrian hospice for great ice coffee and an amazing view of the city at the roof. Venture out of the old city and visit the artists village with really beautifully decorated houses, and colourful houses all around! A really picturesque place!

I’ll recommend the Mount of Olives tour with Sandeman’s. They provide transport up the hill, and the guide was very informative and could answer all our questions. If you are looking for something less common, I took a really cool photography tour with Mekomy. It was a very unique experience, and I definitely got to see Jerusalem in a different light.

Limiting most of my remarks to Jerusalem, since you indicated this is where you're visiting.

Sleeping at a non-cloistered French-run nunnery in the Christian Quarter may be a different lodging experience. The nuns are all Arab, except Mother Superiore; they speak Arab and for us tourists also French and some English. Decent French breakfast, as I recall.

Early in the morning or later in the evening, go to the excavated structures that are south of the Temple site, go inside one of the structures, and just sit on the floor and contemplate the last three thousand years of history at that place.

Early in the morning by yourselves (not with a tour) visit the Church of the Holy Sepulchre, before others get there. Possibly this wouldn't be of great interest if you're Jewish (although He was, too).

If you're Christians, you may want to visit the Galilee area, which is a good bit north of Jerusalem, of course.

There are about a hundred (or a thousand) other "more unique and interesting things to do" in Jerusalem, aside from the typical tourist attractions. Not enough room here to even start to list them all. This is a place where you might want to hire a guide.

A high point for me was the Chagall windows in the Jerusalem hospital. Amazing!!

When exactly are you going to be in Israel? October has holidays that will greatly impact sightseeing. Yom Kippur begins on Friday Oct 3 before sundown and ends on Saturday Oct 4 after sundown. The country virtually shuts down for about 26-28 hours - no shops, no public transportation, no taxis, no restaurants, no amusements, except in the Arab neighborhoods of Jerusalem. Then Oct. 9 through Oct 16 is the Festival of Booths (Succoth). The 9th and 16th are holidays, shops and many restaurants and some sights are closed, no buses or trains, but there are taxis (similar to every Shabbat). On the intermediate days schools are closed, many businesses are closed, and about 1/2 the locals are on vacation and going to all the attractions you may be considering. In other words, everything - especially in the Old City - will be crowded with locals as well as tourists.

Natalie wrote visit the western wall during shabbat and watch the singing and dancing happening then This happens every Friday just after sundown. You are welcome to watch (please do not try to take any photos - there is a religious injunction againt it - or join in. Men and women are in separate sections so decide in advance when and where you will meet up.

Kent is so right about going to the Holy Sepulchre Church early in the morning. I don't know what time they open (7? 8? maybe?). It is not a church like any you've ever been to. So you will need a guide or a guidebook to explain it to you. It is a mosaic of Christian religions and architecture. There is a burial cave that goes back to the time of Jesus. There are excavations of chapels that are 1500 years old, and graffiti from Crusader times. A few meters from the Holy Sepulchre is the Lutheran Church of the Redeemer. Go in to the sanctuary which was built a little over 100 years ago, but according to plans of a Crusader church. For a shekel, you can use their bathrooms (the best in the Old City). For a few shekels you can climb the bell tower for one of the best views in the Old City.

There is a lot to explore in the Old City. One of the very best is to take the tour of the Western Wall http://english.thekotel.org/VisitorInfo.asp?id=1 You have to book in advance since there are only a few in English each day and they are very popular. Just wandering through the streets can be fun once you get away from the main shopping street.

The Israel Museum has some great exhibits. The Archaeology Wing is very well done, so is the Judaica Wing (including period rooms and traditional costumes from different countries). There's also a model of the Old City just before the destruction of the Second Temple, or just after Jesus's time.

The Mahane Yehuda market is another of my favorites. It is closed on Shabbat, pretty dead on Sunday. Wednesday and Thursday are the big shopping days, and Friday (until mid-afternoon when it closes down) is for the last-minute shoppers.

Isn't there an old joke similar to the Subject line, whose punch-line is, "Oh, she moved to Tel Aviv!"

I am not sure what you mean by adventure but probably one of my favorite visits was to Yad Vashem the Holocaust museum and memorial. I was only there a few hours you could spend hours viewing everything. Warning if you go.....the Children's museum will leave you speechless and probably in tears by the time you exit. I know it did me, I had to wonder the Avenue of the Righteous just to ponder what I witnessed.
Ben Yehuda street was also a favorite of mine for a night visit.