Clear the E.G.G. Station Zone without taking any damage.
This is by a long way the hardest achievement in the game, but is much easier than it's XBLA counterpart. This is due to the fact that if you have just died or been hit, you can quickly hit the home button on your phone, then restart (make sure you open the app/game again), and you will be asked if you want to continue from your most recent checkpoint. This means you will carry on at the stage you were just on, meaning if you are on the final boss, you don't have to spend 4 or 5 minutes getting back there again.
First four forms
The first four forms are slightly weakened versions of the bosses from the first four acts. They all require just four hits to defeat.
Stay on the left platform, and jump and homing attack four times quickly, you should be able to deal with him before he gets his special away.
Enter the area slowly, and land on the left platform before the flipper. Jump up, and you will get at least two free hits on him. On the way back up the right hand side of the half pipe after this, jump over and hit him, then homing jump and he should be done.
When you enter the area, stay on the far right, and when the column is on the way up, jump and homing attack him for a free hit. Then, you need to learn the patterns and hit him when you can. Note that although you can normally only get one hit to register per pattern, if you jump when the level is reverting back to normal, and homing attack, you can sometimes get a second hit to register.
Take your time, and watch out for the mines he drops, as it is easy to be focussed too much on the bouncing bombs. As soon as you are in range, attack him. On the WP version, this is probably the hardest of the first four forms, but keep on abusing the home screen trick until you get past it!
This is where the real challenge begins. First of all, keep jumping at him. Homing attack if you want, but this can be risky, as sometimes he will fire his arms at you. If you are charging him at this time, you can easily get hit by these, so be careful. Every now and then, or if you stay too far away, he will fly up, and home into you in an attempt to land on you. Try and get him as close to the edge as you can, and move away at the last minute (the sound effects really help, so if you can, have them on). Once you have damaged him enough, he will stop and start to smoke and electrify. DO NOT ATTACK HIM.
Go to the edge of the screen, and charge you spin dash. He will run towards you, but if you release your spin dash you can run under him and to the other side. He will then fly up and try and crush you again, so go to either side and move out of the way at the last minute and go to the other side. He will then fire his arms at you, which will then float back down. Jump on this three times and it will fire back at him, at which time you can jump on him, followed by a homing attack, twice. He will then repeat the process. You need to do this four times, but be warned. Sometimes he will do his run thing towards you, where you have to spin charge under him, and the last couple of times he will fire pink projectiles randomly too, which you need to avoid.
Eventually, he will fly up one last time, and the cursor showing where he will land will go from side to side. Make sure he isn't gonna land on you, and jump just before he hits the ground, else he will stun you and often cause you to fail. Hit him one more time to defeat him for good!
A video for this can be found below, credit to the original uploader.
Just earned the Achievement for myself using a little trick I found earlier in the day. The game has a "resume" feature that lets you start from your last Star Post if you exit the game with the "Windows" button. I happened upon it accidentally...I thought I lost my progress in the level when I hit the button, but when I went back to the game, it asked if I wanted to resume from my latest Star Post checkpoint, so I said "OK." While playing the game, I realized I might be able to use this feature to get Untouchable.
Anyway, long story short...it works. You still have to beat each Robotnik form without getting hit (including that annoying, final one), but you don't have to do it all in one run per se. If you get hit fighting a particular form, just hit the Windows button go back into the
Argh! Post got cut off. Basically, whenever you get hit, press the Windows button, go back to the game and choose to start from your latest Star Post. You still have to beat each form (including the last one) without getting hit, but this makes it a little easier as you get to retry without doing each form all over again.