Forza Horizon

The game and DLC have been delisted from the Marketplace as of October 20th, 2016. If you bought the DLC previously, this achievement should still be available for you. Otherwise, there is no way to earn it as there are only 5 Nissans in the game.

The original solution is retained below for future reference.
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Here is a list of Nissan cars available on disc. Note that there are only 5 on disc.

1969 Fairlady Z 432
1971 Skyline 2000GT-R
2000 Silvia spec-R
2002 Skyline GT-R V-spec II
2012 GT-R Black Edition

In addition to those above, there are 4 additional DLC cars that can be used for this achievement.

1970 Datsun 510 (February Car Pack)
1993 240SX SE (November Car Pack. it's recommended you get this pack as it comes with additional required cars)
2010 370Z (Pre-Order Car Pack)
1998 R390 (this one is free!)

Also, the R390 is worth 1,200,000 Cr in-game, which contributes to the "It's Just Money" achievement.

Remember that car club cars do not contribute to this achievement. You must own the cars to earn a medal for them. You can sell the car after earning the medal for it.

Please comment if I missed something or can improve this solution in any way.

You need to earn at least one medal in 7 different Nissan cars. Press up on the D-pad to see the requirements for the medals.

Beware, although some cars are free to download, they still need to be purchased in game with either in-game credits or real money (MS Points), but that is the case for every car.

Unicorn cars
I do not include "unicorn" cars as free as you need to earn them and they are extremely rare, so most gamers will not have access to them.

Barn discovery cars
Barn discovery cars are available in the main game, but you have to progress through the story and multiplayer, to have them appearing on your map. Then you have to find them in the barn (wrecked). You have to wait until they are fixed before you can use them. There are 9 Barn discovery cars.
* 1964 Aston Martin DB5
* 1958 Aston Martin DBR1
* 1981 BMW M1
* 1992 Bugatti EB110 SS
* 1957 Ferrari 250 Testa Rossa
* 1956 Jaguar D-Type
* 1954 Mercedes 300SL Gullwing Coupe
* 1971 Plymouth Cuda 426 HEMI
* 1965 Shelby Cobra Daytona Coupe

Loyalty cars
To get your loyalty cars you need at least a save file from either FM3 or FM4. Depending on your level, you get those cars:
* 2011 MINI Cooper S (Level 1)
* 2011 Citroen DS3 Racing (Level 5)
* 2012 Dodge Charger SRT8 (Level 10)
* 1969 Chevrolet Camaro SS (Level 15)
* 2008 BMW M3 E92 (Level 20)
* 2012 Jaguar XKR-S (Level 30)
* 2009 Gumpert Apollo S (Level 40)
* 2012 Lamborghini Aventador LP700-4 (Level 50)

Forza Rewards Program
https://rewards.forzamotorsport.net/
Depending on your tier, you get rewards monthly. Redeem them on the site and they will be deliverer in your game within 4 hours. The rewards can be money, cars, tokens,...

Available DLC Packs
* October Car Pack (1957 Chevrolet Bel Air - FREE)
* November Bondurant Car pack (2012 Jeep Wrangler Rubicon - FREE)
* Season pass (2006 Lamborghini Miura Concepts - FREE)
* December IGN Car Pack (1971 AMC Javelin AMX - FREE)
* Rally expansion pack NOT FREE - NO FREE CARS
* January Recaro Car Pack (1995 Ford Mustang Cobra R - FREE)
* Honda Challenge Car Pack - FREE pack, includes 3 free Honda Cars (2013 Civic Si Coupe - 1986 Honda Civic Si - 2013 Honda HDP Fit - FREE)
* February Jalopnik Car Pack (1966 Ford Country Squire - FREE)
* March Meguiar's Car Pack (1998 Nissan R390 - FREE)
* April TopGear Car Pack (2002 BMW M3 GTR - FREE)

Complete List
* 1969 Nissan Fairlady Z 432 (main game)
* 1970 Nissan Datsun 510 (DLC)
* 1971 Nissan Skyline 2000GT-R (main game)
* 1993 Nissan 240 SX SE (DLC)
* 1998 Nissan R390 (FREE DLC)
* 2000 Nissan Silvia spec-R (main game)
* 2002 Nissan Skyline GT-R V-spec II (main game)
* 2010 Nissan 370Z (DLC)
* 2012 Nissan GT-R Black Edition (main game)

Summary: 5 cars (main game) + 3 cars (non free DLC) + 1 car (free DLC)

I suggest you buy the November Bondurant Car Pack as you need to buy it anyway if you want to unlock the next achievement: