Speakers like the JBL Partybox 310 although they will play deeper just won’t have the output needed for larger areas especially outdoors, so these are the best suggestions for extremely loud speakers.
If you absolutely can’t live without an internal battery and need the loudest speaker possible, you’ll be limited to the Teufel Rockster (EU only) or the Soundboks Gen 3/4 (126db peak). Here they are compared
Also remember that you can get a power station for any speaker mentioned here. See this page for more information about Power Stations
The best portable speakers without a battery are generally the QSC K12.2 (132db peak) or the EV EKX-12P (132db peak). These speakers will by far sound the best and get extremely loud while still being portable and light enough to transport easily. You can also find these speakers open box or refurbished if you don’t have the budget for a new one.
If you’re looking for a cheaper price point under $500 check out the Yamaha DBR12 (131db peak) or EV ZLX12BT/ZLX12P (126db peak). They are about as cheap as you’ll want to go while still sounding good and being reliable.
There are even lower end brands and models, but they definitely won’t sound as good, get as loud and possibly will just shut down and overheat when you need them the most.
If you’re just using one speaker you can likely get away without a mixer as they have mic and line inputs, but when you get serious with them you’ll want a mixer to be able to send stereo output to a pair or more of them. A good quality mixer like the A&H zed10fx talked about here will have better reliability and good preamps. For something cheaper the Behringer Xenyx 802S is reasonable.
There are models of these speakers with 15” woofers also which will reach down a bit lower (~45hz at a 10db rolloff) but the tradeoff will be some frequency response issues around the crossover point (~1.7hz) as the bigger woofer will have frequency response issues playing up that high. The 12” woofer versions of these speakers will generally have the best overall frequency response.
If you need bass deeper than 45hz at a 10db rolloff you’ll likely want to add on PA subs like the QSC KS118 or EV EKX-18SP. These will generally give your tops cleaner sound and more power as they don’t have to drive the deeper bass along with getting you deeper bass down to around 35hz at a 10db rolloff.
Some reasonable cheaper subwoofer options are the EV ELX200-12SP or even cheaper the Behringer Eurolive B1200D-PRO. Keep in mind that these cheaper 12” subs will not have as high of output as the bigger 18” subs.
A lot of this information exists in a few places in the r/livesound sub. Here’s a few great links there that I’ll add to as I find them:
Whats the hierarchy of speakers?
What Active PA speakers to buy?!
Active Speaker recommendations
Looking for recommendations on good powered speakers for live shows