These are probably my favorite tests, off the top of my head:
West Indies vs Australia at Kingston, 1995
The game where Australia stopped WI's unbeaten series record and became unofficial world champions. Steve Waugh played one of his greatest innings (if not the greatest) and endured a barrage of bouncers the likes of which Alan Border said he'd never seen before. Also spawned one of the greatest dummy spits, when Richie Richardson claimed it was the weakest Australian side he'd ever played against.
India vs Australia at Chennai, 2001
After the horrendous wake-up call at Kolkata, I thought it was an aberration, and Australia would still take the series. Aargh, how wrong I was. We looked pretty out of the match when Harbhajan ran through us yet again, but at 8/130-odd, suddenly we looked a chance again. I often wonder about how things might have been so different if Dighe hadn't survived that huge LBW appeal. Just a great test match all round, and the noise level towards the end was absolutely immense.
South Africa vs Australia at Port Elizabeth, 1997
A curiously topsy-turvy match in which the advantage swung dramatically to and fro throughout. Mark Waugh played what he called his best innings ever in conditions in which the bowlers prospered, and Ian Healy, concerned that he might run out of partners, ended the match with two wickets in hand by hitting a four and a six to win the match for Australia.
England vs Australia at Birmingham, 2005
We all lived through this one fairly recently.
Australia vs Pakistan at Hobart, 1999
After the teams were fairly even when their first innings' were completed, Pakistan, with the aid of a century from Inzamam set Australia an improbable 369 chase for victory. At 5/126, a predictable Pakistan victory was on the cards, but Gilchrist, in only his third test innings, gave us a preview of things to come, smacking 149 off 163 balls, and making the Pakistani bowlers look panicky and clueless as to how to stop him. With Langer, who also played an excellent innings, he took Australia all the way home and Pakistan failed to even get into the tail after removing the top order so efficiently (although, to be fair, Langer was given not-out to a thick edge to the keeper on 76).
Australia vs New Zealand at Melbourne, 1987
Australia, thanks to a 90 from Peter Sleep (NZ had awful headaches with the tail in both innings) held a 40 run lead after the first innings, and then were set 247 for victory. But NZ fought back after Australia were looking reasonably comfortable at 5/209, and with six overs to go, Australia were 9 down for 227. Whitney and McDermott decided survival was the best option (they only had to draw to win the series), and Hadlee, who was level at the time with Ian Botham's 383 test wickets, couldn't break the pair, and they hung on for a nail-biting draw, 17 runs away from victory.
Australia vs South Africa at Adelaide, 1998
On a docile pitch, SAF amassed 517. Mark Taylor carried his bat through Australia's entire innings to crucially avert the follow-on, while Shaun Pollock ripped through the rest of the Aussie bats to take 7/87 in 40+ overs. On the last day, Mark Waugh, who was dropped four times, made a century and held out South Africa's charge for victory. There was some controversy towards the end of the match concerning a very late not-out decision on a hit-wicket appeal, because the Saffies didn't understand the rules. This resulted in another dummy spit, this time from Cronje, who allegedly threw a stump through the door of the Australian dressing-room.
(For some reason these dummy-spits from opposition captains really appeal to me, so I can see why England supporters so enjoyed Ponting doing his block during the Ashes
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