How can I check that a given anime is really over? I have been frustrated recently when I watch the last existing episode of a given anime only to find out that the story is not over yet, and I will have to wait for another season (and who knows when that would be). Examples:


No Game No Life
Ao no Exorcist
Made in Abyss
Gakusen Toshi Asterisk


Checking No Game No Life in MyAnimeList, for example, I can see:


  
  Status: Finished Airing
  No other anime listed as "sequel"
  


I went to watch it, thinking it would be a full story in those 12 episodes, but guess what? The story is not over at all. Now here I am waiting for another season, that I don't even know when will come, if at all...

I know that whether a next season will be produced or not can't be predicted. The producers might decide it's not worth it for whatever reason. But some animes simply frustratingly end in the middle of the story. I was hoping that there was a place I could search for this information (if the anime is currently completed or "paused/canceled" in the middle of it). For example, I wouldn't have watched No Game No Life if I knew beforehand it was like that.

Since MAL's status of "Status: Finished Airing" is not what I want, how can I check that an anime is really over?

Notes: for my purposes, animes like Code Geass and Steins;Gate are to be considered complete, even though both of them have a new season coming, the already existing story is 100% complete (of course I only know this because I already watched them).
My suggestion would be to look if there is source material. While I am not 100% sure, I think all the anime you list in the first list are based off the manga.

This process would involve a lot of wild speculation as such it may or may not be accurate and as such would only be served to guide.

To start using No Game No Life as an example, look on Wikipedia to see what it's first media form was, in this case 


  No Game No Life (Japanese: ノーゲーム・ノーライフ Hepburn: Nōgēmu Nōraifu) is a light novel series by Yū Kamiya.


as such it's very likely it would be following the Light Novels. Next, in the infobox on the right, look for the "Original run". If it's officially finished, it would have listed an end date. In this case, the Light Novel has


  April 25, 2012 – present


However, if possible, see also if you can find another source. Baka Updates Manga has info on the Light Novel and under "Status in Country of Origin" it says


  9 Volumes (Ongoing) + 1 Gaiden


Conclusion: No Game No Life's anime will not be the end of the series as its source material is still ongoing.



However, what happens if the source material is complete? Or you want to know if the anime is up to date with the source material? Well, this is simply looking up to the Anime airing dates to the source material's release dates. While not accurate, since anime takes time to produce and the source material could be coming out while it's being developed, it can help you see how much of the story isn't going to be told.

Using No Game No Life as an example, we see the anime's run-time was


  April 9, 2014 – June 25, 2014


looking at the list of Light Novel Volumes we see that Volume 6 came out April 25, 2014, the only volume to come out in 2014. Since it also came out after the anime started airing, we can assume that there won't be any content of Volume 6 in it.

But can we narrow this down expecting 5 volumes of the light novel? Yes we can. For No Game No Life, Wikipedia has information on the anime


  On July 27, 2013, Monthly Comic Alive announced the anime adaptation for No Game No Life was green lit. It is directed by Atsuko Ishizuka and animated by Madhouse. The series premiered on April 9, 2014 on AT-X;


going back to our volume list for the Light Novel, it seems we can rule out Volume 5 as it was released after the announcement of the anime, so now we have 4 volumes.

Conclusion: No Game No Life's anime at most should cover 4 out of 9 volumes that have been released to date.

Keep in mind however that the story content might be less, or it might deviate from the source material (e.g. the 2003 series of Full Metal Alchemist and fillers in long running anime)



Another title you mention in your question is Steins;Gate. It's a bit more complicated because its source material is a Visual Novel with multiple endings and routes. These tend to be a bit more complicated as sometimes an Anime inspired from a Visual Novel with multiple endings will only adapt one.

There are other times the source Visual Novel (or game) is linear, like Tears to Tiara and Utawarerumono. But if there are multiple games, it will only adapt one game, i.e. currently the Tears to Tiara anime only adapts the first game (particularly the PS3 release) and not the second game while there are multiple Utawarerumono anime as there are Utawarerumono games (some of the later games I think were inspired by the more recent anime. However, I'm not 100% sure)

However, sometimes this is made easy like with Fate/Stay Night with each route having been named and the more recently anime being named the same, such as Fate/Stay Night: Unlimited Blade Works by Ufotable. We can assume it follows the Unlimited Blade Works route of the game while the upcoming Fate/Stay Night: Heaven's Feel movies, also by Ufotable, will follow the Heaven's Feel Route. While this won't tell you which ending they may choose, knowing that there are 3 routes in Fate/Stay Night, you can get a rough guess that each of these is really just 1/3 of the whole story you have in the game.



The above however only works if your anime is based off a source material like a Light Novel, Manga or Game and can still be inaccurate without knowing the content of the source martial and the anime. 

You can, however, ask a question here. For instance, I asked if Spice and Wolf followed the manga and learned that it actually follows the Light Novel and it adapts 1,2,3 and 5, skipping Volume 4.

for original anime like Code Geass, you can assume that it has finished. However, there are times where you can ask a question if it does continue but not asking when it'll come out. For Code Geass, while there was no R3 manga, there was Jet Black Renya and Akito of the Exiled manga which, while not R3, did expand on the Code Geass Universe.



In the end, this comes down to doing research on the release, airing and announcement dates, and making assumptions based on these and asking here if you want a definite answer on things like how loyal an anime follows its source material, exactly where it adapts to and which are filler content.