>>655
Bishonen Valet. Even ignoring rules lawyering about the exact meaning of "impossible" that clearly ignores what the image is trying to communicate, the Bishonen is by far the most helpful. The Loli and Imp do the same thing but with an ending condition that could possibly get met accidentally, and the demon servant's most useful feature comes with a big drawback.
The strange object choice is less straightforward, but only because the gemstones and the die are both decent options. The jail card is useless because it only covers petty theft and the SupermoonMedia account provides a paltry $3000 annually, which is nothing compared to what you can get with the die or the value of the gemstones.
With the gemstones, you should be able to safely keep two or three electronic devices running indefinitely, you will probably be "resistant" to poison or rotten food at any given time (although since it's not said how resistant it really isn't clear how helpful this is), and about once every three months you get super luck. People have already mentioned how an extra 10-15% odds of winning can push the odds out of the house's favor and into yours when gambling with the die (although note that +15% odds of winning the lottery is not at all the same as 15% odds of winning the lottery). Ultimately I'd go with Dictator's Darling. I think I can benefit more from occasional days of superluck than from trying to squeeze a living out of gambling games that I'm something like 55-60% likely to win. I'll lose nearly as much as I win.
The pets are all pretty decent choices. None of them are very overwhelming but all of them are pretty neat. The rat seems like the most helpful.