Li-Ion works best. Some new Li-Ion 9Volt batteries can be recharged by USB. You do nit need to buy a battery charger. These are the best ones to get.
Beware that some of the new 9V batteries with the highest mAh rating are also larger than the ASTM standard allows for 9V batteries, so they do not fit well into the A9 (and many other devices… these batteries are just too large).
Also, upon testing, I find that there is almost no correlation between the advertised energy capacity of a battery compared to the actual measured battery capacity. Translation: many battery companies lie about it. After all, how is the average customer going to check?
I suggest you buy a 9V battery that has been tested by someone on this forum and fits well into the A9, or that you test a few different ones for yourself, realizing that some will just not fit well. Alternatively, I suggest buying batteries with mAh ratings in the middle of the believable range (~ 650 mAh).
Also, keep in mind that battery companies do regularly change their products, so a battery brand that might have fit well last year might not fit so well this year, and vice versa.