Silverrails there actually is a rarely used arcane feature of the powerful Nikon SB-800 which will allow you to control it with your Nikon D3300's built in flash. It is called SU-4 mode
If you haven't yet downloaded the SB-800 User's Manual I strongly recommend you do so.
https://download.nikonimglib.com/archive2/9nBAS00MAAL201vA1P470ePJuS88/SB-800_EU(En)03.pdf
That said I've learned from this forum that precious few seem to read through a 137 page manual (though it is stellar!) To this end I've provided a step by step guide on how to setup your SB-800 in SU-4 Mode.
This will empower you to start using your SB-800 as a remote today absolutely free!
Albeit as a legacy wireless slave.
How To Slave an SB-800
1. First, turn on your flash.
2. Now get into the submenu by pressing and holding the "sel" button for 2 seconds.
3. Arrow right (the "single tree" button) to get the upper right quadrant highlighted.
4. Push the select button again.
5. Scroll down ("-" button) until "SU-4" is highlighted.
6. Push "sel" again to select.
7. Push and hold "sel" for 2 seconds to get out of the submenu.
8. Your flash will now say "REMOTE."
You are now in SU-4 mode Silverrails, a legacy, wireless, slave mode for which the SB-800 is backward compatible to any flash on camera or otherwise (But not for TTL). By pressing the "mode" button you can toggle between automatic and manual. Choose manual.
By pressing the "+" or "-" buttons, you can change power in 1/3-stop increments all the way to 1/128 power.
Important: You want to use all flashes in the manual mode, so they will not influence each other's output.
The "eye" of the slave is the circular optical port on the left side of the lash if you are facing the front of the flash. So you will want to rotate the flash body so the window faces the master flash.
Caveat: Please be certain to set the D3300 on-camera flash to manual and avoid TTL preflashes.
The SB-800 in SU-4 mode is so sensitive it may fire from those little preflashes.
Ok, have I used the SU-4 mode? Absolutely! I have a spare SB-700 that I have set to SU-4 Mode that I use in my studio when I need a touch of fill for full length catalog fashion. I place it on its stand on the floor at ~ 8 +/- feet out and it provides just enough fill to illuminate the talent’s lower wardrobe. It’s a handy quick fix during a long catalog session. Those pesky designers want their textile artistry clearly display meaning no deep shadow areas.
Hope this helps Silverrails and saves you the costs and frustration in using cheap third party solution’s whose documentation can be somewhat arcane.
Oh as an extra benefit since there are no complex signals between the controller and slave this setup can have a range of well over 100 feet! How cool is that!