I have sifted through the YouTube offerings for you and found a decent selection of old Mecha-ike clips.
This guy, this guy, and this guy have a ton of videos.
爆走数取団 – Counting Biker Gang
One game they don’t play anymore is called 爆走数取団 (ばくそうかずとりだん). This was my favorite game for a long, long time. The male members of the cast dress up like biker thugs and play a counting game. The first person says an object, for example "books," and the next person has to use the appropriate counter to count that object, so 一冊. He would then say a different object, say "CDs," and the following person would count that, 二枚. This continues until someone makes a mistake, at which point they have to fight a sumo wrestler. Sweet game. I remember seeing part of this episode where Akiko Wada cried: 1, 2, 3, 4, 5.
And here is the always unpredictable Ega-chan versus the sumo: 1, 2.
色取り忍者 – Irotori Ninja
They have replaced 数取り with 色取り. Here are three clips I made introducing the 色取り忍者 (Irotori Ninja) Game and showing how it can be played at elementary school: 1, 2, 3. Also, a funny clip introducing the guest ninjas for that episode.
オッファー – Offers
Once a year Okamura and Yabe both get "offers" where they do other jobs. Yabe has worked at an onsen, in a pastry shop, and as a manager for a fashion model among other things. Okamura has performed with SMAP and EXILE. In the fourth offer, he performed with a Chinese juggling troupe: 1, 2, 3 … this goes on to 13, so you can find the rest yourselves. And here’s are clips from Okamura’s offer with EXILE, which was hugely popular: 1, 2.
抜き打ちテスト – Pop Quiz
One of the other famous bits they do is the 抜き打ちテスト（ぬきうちテスト, pop quiz), where they surprise all the participants by giving them a test. Then they make fun of the stupid answers and crown the バカ of the group. They have played this game with numerous different groups, most recently with イケメン guys. Here is the first one they did: 1, 2, 3, 4.
もっとも受ける芸人は誰！ – Who is the Funniest Comedian!?
I don’t think I’ve seen this kind of episode live, but it’s called もっとも受ける芸人は誰！(もっともうけるげいにんはだれ!). All the members get a chance to make a target audience laugh. There are four different episodes, each divided into several videos:
1: 1, 2, 3, 4
2: 1, 2, 3, 4, 5
3: 1, 2, 3, 4, 5
4: 1, 2, 3, 4
This game is making fun of a game played by the manzai group とんねるず on one of their shows. They invite two celebrities and have them eat several dishes, one of which they hate. The other celebrity then has to guess which one the other didn’t like. This is similar, but they have to guess which comedian they didn’t like: 1, 2, 3, 4, 5.
油谷さん – Mr. Oil-Valley
Here is Yamamoto’s famous 油谷さん (Aburatani-san, Mr. Oil-Valley). He would grease himself up and surprise a male staff member on their birthday or near the birth of a child: 0, 1, 2, 3, 4, 5, 6, 7, 8(*), 9 (These clips are a little frightening knowing that he was kicked off the show in 2007 after being accused of statutory rape. The charges were dropped after reaching a settlement, but not before he lost all his regular appearances and contract with Yoshimoto Kogyo. I haven’t researched it at all, but I think they stopped playing the motorcycle counting game once he left. Ah, a quick check on wikipedia shows that I’m right.)
* I vividly remember learning the meaning of オギャー from this clip. オギャー is the sound a baby makes when it is born.