Sonic the Hedgehog 4: Episode I

There's an easy/sneaky way to nail this achievement down. There is the 2nd Casino Road Act, which contains a groovy little card game. As you fly by, little icons appear at the bottom of your screen. Nab 3 of a kind of coins to get a 10 coin bonus, and 3 of a kind of Sonic to nab an extra life. These can really add up if you get 6 or more in quick succession. (Not to mention the "extra life" music plays quite a bit, hopefully it doesn't annoy you)

Depending on how well you do and how well you can hold your rings, you can nab 10-15 lives in one session on this zone. If you fall down a pit, before you hit the bottom, pause and hit "Retry". It will take you to the start of the level, and you will still have the lives you have earned.

Keep doing this about 5 or 6 times, and you'll have 25G on your gamerscore.
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Have I missed anything? If so, let me know in the comments. Cheers. :)

I gotta be honest I got most lives on special stage 1 trying to get that collect all rings and clear achivo. If you get most rings and collect the 1 up that replaces the emerald you get 3-4 lives in about 2 mins effort so easy to farm if you want to get the achivo with no hassle or if a few short from the main game.

The easiest place to grind for lives is Act 2 of Casino Road: Road of Cards. Every time you run past the large cards, you will see the cards collected at the bottom of the screen. If you get 3 of a kind of Sonic, you get a life. There are tons of these cards throughout the level, and after a run of the level, you usually gain around 10 lives.

This achievement came naturally to me while trying to get all 7 chaos emeralds. By the time I got my 7th I had 90 lives, so my suggestion would be to go for that achievement first and mop up in Casino Street Act 2 afterwards. If you're trying to 200 this, might as well kill 2 birds with one stone.

Things slow down too much in the 2nd half of the Casino Street Act 2. Just play the first half of the level (until you got all the cards in the big room full of cards and springs after the 2nd card road), and pause the game and quit. You keep your lives and dont have to waste time with the 2nd half of the level. 7-11 lives in a 3 minute run each time is what I got.

As others have mentioned, Casino Street Act 2 makes this achievement a cinch. However, if you progress through the game as normal -- especially if you're going for other achievements like destroying 1000 enemies -- you may find that this achievement will pop up when you're not expecting it. Lives accumulate very quickly, and unless you suck at Sonic, you won't be losing many of them along the way.

If you just want this achievement out of the way, do as the others have mentioned and run Casino Street Act 2. You can get upwards of 25 lives in a single run, if you're very lucky.