My '99 has almost 150,000 miles, and it's running great. The only problem I had was a hairline crack in the radiator which ended up needing replacement at 146,000 miles. Avalons are excellent highway cars because of how comfortable they are on the highway, they have great power so it doesn't take much effort to pass somebody or merge, pretty damn fuel efficient, and pretty stable and smooth and high speeds.
Things to look for: As others have mentioned, oil sludge is an issue. Personally, mine is fine.
Timing belt should have been replaced at around that mileage.
Suspension may be noisy and pretty soft
Water in doors. Mainly driver's side passenger door.
Radio buttons sometimes fall out