When loading minecraft from a separate disk than the OS, in this case, my Crucial SSD (C Drive) to the minecraft location set in my profile options (G Drive) the game crashes because of an IOException ' CreateProcess error=740, The requested operation requires elevation'. (I should also note that java is also installed on my G Drive). The crash can be seen from the Launcher log but not the game output.

[23:56:31 ERROR]: Couldn't launch game
java.io.IOException: Cannot run program "G:\Program Files (x86)\Minecraft\runtime\jre-x64\1.8.0_25\bin\javaw.exe" (in directory "G:\Users\Aidan\AppData\Roaming\.minecraft"): CreateProcess error=740, The requested operation requires elevation
at java.lang.ProcessBuilder.start(ProcessBuilder.java:1048) ~[?:1.8.0_25]
at com.mojang.launcher.game.process.direct.DirectGameProcessFactory.startGame(DirectGameProcessFactory.java:14) ~[launcher.jar:1.6.11]
at net.minecraft.launcher.game.MinecraftGameRunner.launchGame(MinecraftGameRunner.java:208) [launcher.jar:1.6.11]
at com.mojang.launcher.game.runner.AbstractGameRunner.onDownloadJobFinished(AbstractGameRunner.java:186) [launcher.jar:1.6.11]
at com.mojang.launcher.updater.download.DownloadJob.popAndDownload(DownloadJob.java:122) [launcher.jar:1.6.11]
at com.mojang.launcher.updater.download.DownloadJob.access$000(DownloadJob.java:12) [launcher.jar:1.6.11]
at com.mojang.launcher.updater.download.DownloadJob$1.run(DownloadJob.java:86) [launcher.jar:1.6.11]
at java.util.concurrent.Executors$RunnableAdapter.call(Executors.java:511) [?:1.8.0_25]
at java.util.concurrent.FutureTask.run(FutureTask.java:266) [?:1.8.0_25]
at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1142) [?:1.8.0_25]
at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:617) [?:1.8.0_25]
at java.lang.Thread.run(Thread.java:745) [?:1.8.0_25]
Caused by: java.io.IOException: CreateProcess error=740, The requested operation requires elevation
at java.lang.ProcessImpl.create(Native Method) ~[?:1.8.0_25]
at java.lang.ProcessImpl.<init>(ProcessImpl.java:386) ~[?:1.8.0_25]
at java.lang.ProcessImpl.start(ProcessImpl.java:137) ~[?:1.8.0_25]
at java.lang.ProcessBuilder.start(ProcessBuilder.java:1029) ~[?:1.8.0_25]
... 11 more
[23:56:31 INFO]: Deleting C:\Users\Aidan\AppData\Roaming\.minecraft\versions\1.8.2\1.8.2-natives-50437422910334

The problem can be fixed by running the launcher as an administrator.