Making the Game Harder (Apocalypse)
Superhuman? ...too easy.
No Money Exploits
There are a few shady ways to gain money such as with stripping Stormdogs or messing with the engineering screen. It kinda goes without saying but don't do that.
Real Time Only
For those of you who only use turn-based play, while some consider RT easier, it also involves a lot of thinking and reactions on your part and thus allowing for a more interactive, and in some place harder game play.
Now we're talking. The ability to pause makes RT play much easier, and basically removes the necessity for quick thinking. Without pausing, real-time play really comes into its own in difficulty.
In contrast to the Real Time Only you can play in only Turn-Based Combat, this instantly makes a number of aliens a lot more dangerous, especially if your men have low reactions as the Brainsucker and Popper can run up to your men and Brain-Suck you or explode in your face without your men being able to react like they can in Real Time. Additionally, Security Stations do not function in TB, so base defence missions will be far more difficult and will require a substantial garrison.
No Infiltration Graph
The alien infiltration graph is both rather implausible (how, exactly, are you getting this information?) and incredibly effective; simply comparing the list of organisations being infiltrated against the areas where aliens have been inserted will, in 90% of cases, pinpoint their exact positions and let you exterminate them. So, don't use it. This makes it crucial to shoot down as many of the transports as possible, because if they drop troops you'll be faced with the choice of investigating every nearby building (and taking a relations hit) or waiting for an alert (and risking alien takeover).
No Humans Allowed
You can never use human soldiers in combat. Fire the soldiers you start with. Start recruiting from SELF and the Mutant Alliance ASAP. Both soldier types are fairly uncommon but for an extra challenge you can opt to only use androids or only hybrids.
Senate Funding Only
This is basically in the same area as the Council Income Only in UFO and TFTD. You must rely on the Senate's Funding and selling stuff. Unlike in previous titles a huge chunk of XCOM's funding comes from the Senate, so this mode isn't as tough as it sounds.
Don't Run a Sweatshop
You can not generate income from selling XCOM manufactured goods. Anything that you build must be used or stored. The Bio-Trans comes with extra modules, so to settle any arguments don't sell those either.
Don't Raid Other Organizations
You can not raid organizations for their phat lewt. This will be rough especially in the first few weeks as XCOM starts on a tight budget and raiding is the only other source of income. After a few weeks the Senate's income grows enough that you can manage.
Don't Sell Artifacts
Don't sell any alien tech. Mod all alien artifacts to have zero value if you must, or sell it before it gets a price tag from research. Alien tech does pile up in storage after a while so be sure to blow up bodies and make judicious use of "abandon battle". You don't really need the huge surge of artifact money to beat Apoc in any way.
Don't Sell Anything. ANYTHING.
For an increased Challenge, don't sell anything at all! This will hurt badly when upgrading to the Air Hawk, and you will not be able to manufacture for profit nor refund anything you don't like. Also be prepared to manage the huge amount of JUNK that will pile up in your storage, perhaps by abandoning it in battle! Remember that alien bodies are incinerated and not sold so burn away.
Unlike in previous XCOMs the starting equipment never really becomes obsolete. Arthropods and Skeletoids are featured the entire game and they can be killed with machine guns or autocannons all game long. UFOs can be destroyed with conventional missiles, while top tier UFOs are still manageable with elerium guns and tanks. Megaprimus definitely gives everything you need to beat the alien menace so go give it a try.
Human Tech Only
You can not use any researched alien projects. No disruptors, no shields, no vortex mines and so forth. This mode does not preclude XCOM from building the hybrid craft needed to breach the alien dimension, but you can not equip them with alien tech like shields or disruptor weapons. On the plus side this saves you a lot of trouble having to research them!
The Toxigun is the single most powerful weapon against the alien menace. So don't use it. Even Alien Gas is more fair than the Toxigun.
Okay, Alien Gas isn't really THAT fair. But you already made it through 90% of Apocalypse without using it. You may as well grit down and power through that last stretch, and maybe even kill the Queenspawn to avoid the temptation.
Stun Grenades are good. Like REALLY good. They can KO brainsuckers as they hatch, get you early game shields, or prevent Multiworms from breaching their offspring. Give yourself a challenge and don't use them.
First you restrict your weapon selection and now this! Megapol armor is pretty good but expect a return of the mortality rates you are familiar with in UFO/TFTD.
If it deals explosive damage, don't use it. If you mind control an alien and he has (or is made of) explosives, don't use it. Better make friends with Diablo right away because the only way to cure the alien menace is with Cleansing Flame. This doesn't exclude you from using incendiary weapons or from setting off those massive chain reacting piles of alien explosives because face it, that's going to happen no matter what. But it's a bit tricky to set off explosives without using explosives.
Conventional Craft Only
All XCOM hybrid craft are prohibited. The only exception is to have a single Bio-Trans (One only or fleet cap: 1), which is the absolute minimum to probe the alien dimension, invade it, and beat the game. Good luck invading the alien dimension with one little ship! Note that the aliens send more UFOs to Megaprimus than they can manufacture, so you can slowly empty the alien dimension by killing all invaders.
Hoverbikes are one of the most effective combat vehicles early in the game and they can serve you well into midgame. Sell them off and start using the other vehicles instead. Combine with Conventional Craft Only for some serious endgame trouble.
Hoo boy. Elerium fuels the best conventional craft weapons in the game. Mid tier UFOs will be tricky targets and the final large types will be nigh impossible to take down. Ammo will be tight so you will need a strong combination of both air and ground forces to succeed. Fortunately you do not actually need to destroy all the UFO types to unlock effective Hybrid craft.
There are three Elerium based items for your agents to use, the Megapol Plasma Gun, the Power Sword, and the Marsec Armor chest unit. All three options are respectably useful but are fairly rare as is so there's no problem managing without it.
Making the game SILLIER (and giving yourself a harder time in doing so)
The below are some silly little challenges I gave myself, I enjoyed them and hope you might enjoy them too:
1. Make X-COM active in Public Relations! ... Make everyone love you, and hate the aliens and the Cult Of Sirius. Making them love you can be settled with just money, but making everyone hate the COS and aliens is a BIT more work... EDIT: Turns out this is actually a VERY effective way of playing the game. I made EVERYONE hate the COS by baiting their temple defence forces to chase down my hoverbike, then parked in front of organization buildings and got them to blow up the building. The other organizations all ended up extremely hating the COS. Furthermore, all the organizations now love each other because whenever someone attacks or raids COS, everyone else likes them more. Government won't be thrilled at all that damage to the city though...
2. Make the COS hate the aliens. You would have to keep trying to bait the aliens into shooting up the COS temples. And I'm not sure whether this shift in attitude will actually stick or the game is coded to have COS forgive the aliens. (edit: I just tried this, and even though the aliens demolished 3 COS temples, COS still loves the aliens a lot) (Edit2: Used Apoc'd and made the aliens love everyone EXCEPT Xcom (cause I heard that might bug the game out) and the also HATE Cult Of Sirius. So, the next bombing mission, they bombed the living heck out of a COS temple. ... checked the relations with apoc'd, and COS is still +100 for the aliens. So, as it turns out, the COS will love the aliens NO MATTER WHAT)
3. Advance human technology beyond the aliens' . The alien's tech level is based on your total tactical score. The humans' tech levels are based on what week it is. If you purposely tank your tactical mission scores, you can keep the alien's using nothing but brainsuckers while even the most low tech human organization has advanced to personal shields and devastator cannons.
4. Kill a megaspawn with just machine gun bullets. This probably isn't too hard once you have personal teleporters, in turn based mode, but it's SADISTIC. ... and it's going to use A LOT of bullets... like really really a lot of bullets. (edit: I just tested it, it's only around 90 bullets, on Superhuman)
5. Stun Raids are naughty. Only naughty people do stun raids. ... that's why, we should make the aliens do stun raids. ... Turns out, when you destroy the Control Chamber, the aliens will send every ship they have remaining on a bombing run of the city. ... Well then, what if you invaded the alien dimension and wiped out everything they have except that you spare all of their Escort Ships? ... you know, those purple things that ONLY have stasis bombs? And can't actually do any damage? Hahaha... Edit: Evidently, this DOESN'T work. I left 6 escorts only alive, and blew the control chamber, and the aliens just went into permanent AFK mode. Escorts are probably coded to ONLY be escorts and can't go on bombing missions.