is a build with everything included at ~750$ possible? (monitor, keyboard/mouse, speakers)

Yeah, you can easily build a good gaming PC in that price range.

This will play anything on at least high settings even Crysis 3:
PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($124.99 @ SuperBiiz)
Motherboard: Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard ($49.99 @ B&H)
Memory: Kingston HyperX Fury Black 8GB (1 x 8GB) DDR4-2133 Memory ($45.48 @ OutletPC)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($49.89 @ OutletPC)
Video Card: Sapphire Radeon R9 380X 4GB NITRO Video Card ($233.98 @ Newegg)
Case: Corsair SPEC-01 RED ATX Mid Tower Case ($34.99 @ Micro Center)
Power Supply: EVGA 750W 80+ Bronze Certified Semi-Modular ATX Power Supply ($64.99 @ Amazon)
Case Fan: Corsair Air Series AF140 Red 66.4 CFM 140mm Fan ($13.55 @ OutletPC)
Case Fan: Corsair Air Series AF140 Red 66.4 CFM 140mm Fan ($13.55 @ OutletPC)
Monitor: Acer R240HY bidx 60Hz 23.8" Monitor ($131.99 @ Amazon)
Keyboard: Logitech MK120 Wired Slim Keyboard w/Optical Mouse ($8.99 @ NCIX US)
Mouse: Redragon Centrophorus M601 Wired Optical Mouse ($13.99 @ Amazon)
Speakers: Logitech S120 2.3W 2ch Speakers ($11.99 @ SuperBiiz)
Total: $798.37
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-01-31 19:54 EST-0500

You should be able to handle almost every game out there at medium or high with this PC. With this budget, you can't gt a great mouse and keyboard, but you could get those later. I also figured you might want some sort of color scheme (most people do) so this one is all blue and white. I also wanted to make sure you wouldn't need to worry about upgrading your monitor later on, so this is a pretty nice one. Here is the build:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-6300 3.5GHz 6-Core Processor ($98.89 @ OutletPC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($24.88 @ OutletPC)
Motherboard: Gigabyte GA-78LMT-USB3 Micro ATX AM3+ Motherboard ($50.98 @ Newegg)
Memory: Crucial Ballistix Sport XT 8GB (2 x 4GB) DDR3-1600 Memory ($34.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($46.89 @ OutletPC)
Video Card: EVGA GeForce GTX 950 2GB Superclocked+ ACX 2.0 Video Card ($141.99 @ NCIX US)
Case: Thermaltake Commander MS/I Snow Edition (White/Black) ATX Mid Tower Case ($44.13 @ Mac Mall)
Power Supply: Corsair CX 500W 80+ Bronze Certified Semi-Modular ATX Power Supply ($34.99 @ Newegg)
Optical Drive: Lite-On iHAS124-14 DVD/CD Writer ($12.99 @ Newegg)
Operating System: Microsoft Windows 10 Home OEM (64-bit) ($87.95 @ OutletPC)
Case Fan: Kingwin CF-012LB 40.0 CFM 120mm Fan ($3.99 @ Micro Center)
Case Fan: Kingwin CF-012LB 40.0 CFM 120mm Fan ($3.99 @ Micro Center)
Monitor: Asus VX248H 60Hz 24.0" Monitor ($119.99 @ Newegg)
Keyboard: Logitech K120 Wired Standard Keyboard ($11.99 @ SuperBiiz)
Mouse: Logitech M100 Wired Optical Mouse ($7.89 @ SuperBiiz)
Speakers: Logitech S120 2.3W 2ch Speakers ($11.99 @ SuperBiiz)
Total: $738.52
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-01-31 20:38 EST-0500

I would change the CPU to an FX-6300 which would be a lot better for the price than an i3 I think. Also you only have 1 stick of memory for the 8GB, get 2x4GB for dual channels.

No, FX-6300 is not better than i3 in gaming - maybe BF4 but that is debatable.

Yes, I do have a single stick of memory. Guess why? Because there are only two RAM slots on the motherboards so by getting a single 8GB stick he can upgrade to 16GB later if he needs to.

Why not get a mid atx motherboard then?

I know the price difference is $60 between mine and yours but you are getting an R9 380X in my build and all high quality components.

I see 4 flaws in your build already. If you really want to waste my time to tell them to you, I will:

1) You are getting a 6-core FX with 212 EVO to overclock on an incredibly cheap board that is not even recommended to use with an FX-8320. Overclocking FX-6300 will bring similar strain. That board is really cheap and is not quality at all.

2) No, FX-6300 is NOT better than an i3 + the i3-6100 is Skylake and will perform better than the i3-4370.

3) GTX 950? Seriously? Add $20 in for a 960 or 380.

4) CX power supplies use low quality capacitors and are not suitable for any form of overclocking (haha you thought he was gonna overclock the FX-6300 on this power supply didn't you) nor are they suitable for constant high loads.

5) For $10 more you are getting a higher quality IPS panel that I recommended.

He said not to add an OS.

His budget is $750-800. All I was doing was making the best use of his money.

First of all, if you feel that is a waste of time to give evidence supporting your choices, then maybe you shouldn't be on a site like this. Your job is to give an answer and support it, so it should never be a "waste of your time". So as to your supposed flaws:

1) The Hyper 212 is a great cooler, and it says no where in my post, or in the OP's post that he has any intention or need of overclocking, so I did not design this build to do so. Having a nice quiet cooler, is always a great thing to have anyway.

2) I agree, the FX-6300 is in no way better than the i3-6100, but it does offer way better performance for dollar. Intel is not worth getting until you can spend at least $200 on a processor. AMD FX processors have the ability and have a great price to fill this gap. When you throw in Monitor, keyboard, mouse, speakers and Operating System, this build becomes essentially a $500 budget build. If you look at almost every low budget build out there, They have either an FX-6300 or a high end Pentium. There is a reason, i3's are too expensive for budget PC's.

3) I would have added in a gtx 960 instead, but He said his budget was $750, $800 max. This generally means they want it to be around $750, but if they absolutely have to, they will spend the extra money, only i it is a must. He said he wants to be able to play on Medium settings, this card will do so, so why not save the extra money.

4) Once again, this is essentially a $500 budget build. Best to save money in this regard and he can upgrade it later if he plans on getting stuff to upgrade. Also, I once again, never said anything about Overclocking, nor did the OP. And constant high loads? This has a total TDP less than 300W, their is no high load on this build.

5) Quality monitor for a cheap price, I don't see what your problem is here.

6) He never said to not include the OS, he said to include everything which usually means that you have to buy an OS and dvd drive as well. If he does need the OS, then your build goes down the toilet. Maybe next time you will read the entire thread before posting this junk.

7) As I stated earlier he wants to keep it around $750. He doesn't want to spend $800, but he will go up if he absolutely must in order to get the performance he wants. If he can get the performance at $750, then that is what he will end up going with.