(Hudson's) Adventure Island

continue: There is a hidden egg on the ledge just before the end of the
level. Open it, take the bee inside, and when you die, hold Right and hit 
Start to continue where you left off.

A more-or-less complete listing of bonus rounds:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1-1, section 3: Stand between the fourth and fifth totem poles and wait for
the platform to appear.

1-3, section 2: Look for the key egg right on the ledge just before the
dropping platform. Once you have the key the platform lifts you to a
bonus level instead of dropping.

2-3, section 2: The key egg is on the stone section right after the "2" sign.
After that, the third platform lifts you to the bonus level.

3-1, section 2: pass the two frogs and jump down two more steps to a long
flat area. Run around and toss your weapon around and eventually the 
platform leading to the bonus area will appear.

4-3, section 4: Jump to the third ice-tower, and jump to find the key egg.
The next moving platform takes you to the bonus area.

5-1, section 1: The key egg is just before the cloud platforms, and the
last cloud platform then takes you up to the bonus area.

6-2, section 1: Jump over the rolling boulder and then wait around for a
little bit for the bonus platform to appear.

8-1, section 3: Stand in the area between the rock with the hidden pot and
the egg with the eggplant inside.