[Fire HD 6 and 7 General] bibikalka:
OK, here is a rough quick guide for spring cleaning. Note that Amazon never intended for us to be moving stuff around, but that does not mean it should not be done! The main objective is to free up some space in /system, so that one can add GAPPS and block updates if the new GAPPS don't like the older Google Play Services. For reference, /system is about 1.1Gb, /cache takes 0.85Gb, and then /data is ~5Gb . There seem to be ~1Gb missing somewhere.

Assuming your adb with root is all go (non-bricking safety first!), go grab the 4.5.3 ROM at Amazon for the unlimited supply of stock Amazon files in case you want to restore something you've messed up (assuming you are on 4.5.3):

https://kindle-fire-updates.s3.amazonaws.com/update-kindle-20.4.5.3_user_453011120.bin

(grab other ROM versions if you need them : http://xdaforums.com/showpost.php?p=61005379&postcount=3

Renaming *.bin to *.zip will enable you to unpack it, and dig out any files you want for restoration.

Not recommended, but an easy hit is "Ivona TTS", you can delete \system\tts\ivona\eng.usa.salli_31536000000\vox* for extra 82Mb (sorry for wrong slashes), if you miss Ivona later, restore from ROM file. But given the next option, perhaps keep Ivona for now since you may gain sufficient space in /system by just moving apps around.

Other easy hits, move these non-odexed apk's from /system/priv-app to /data/app :

06/27/2014 11:56 PM 28,288,129 com.amazon.weather.apk

06/27/2014 11:56 PM 22,233,756 com.amazon.cloud9.apk

06/27/2014 11:56 PM 20,928,116 moffice_6.0.1_default_en00105_multidex_195423.apk

06/27/2014 11:56 PM 16,454,204 com.amazon.kindle.apk

06/27/2014 11:56 PM 16,243,494 com.amazon.photos.apk

06/27/2014 11:56 PM 10,437,348 com.amazon.mp3.apk

06/27/2014 11:56 PM 9,709,408 com.android.email.apk

06/27/2014 11:56 PM 8,719,653 com.amazon.tahoe.apk

06/27/2014 11:56 PM 8,520,630 com.amazon.kcp.tutorial.apk

06/27/2014 11:56 PM 7,829,227 com.audible.application.kindle.apk

06/27/2014 11:56 PM 7,425,248 com.amazon.windowshop.apk

06/27/2014 11:56 PM 7,420,412 com.amazon.geo.client.maps.apk

06/27/2014 11:56 PM 7,038,811 com.amazon.avod.apk

After reboot, they should happily function from /data/app, cleaning up 163Mb of space . There are more things in /system/priv-app that can be moved, but the file size gets smaller, so I did not bother to investigate other ones. moffice also lives in Google Play Store, so you may junk it fully since it's so easy to re-acquire.

These non-odexed apps I did not touch yet (132Mb of space), but they are candidates too if one wants to automate the relocation with an updated Jmz Fire tool. Basically, any benign user app where you recognize the icon is safe to move to /data/app (keep the otter swimming in /system/priv-app just to keep factory reset happy, since I am not sure how it is going to like the moved apk's):

06/27/2014 11:56 PM 14,968,806 com.amazon.kindle.otter.apk

06/27/2014 11:56 PM 12,156,654 HPLegacyPlugin_AmzPreload-release.apk

06/27/2014 11:56 PM 7,665,090 Camera.apk

06/27/2014 11:56 PM 5,654,600 com.amazon.venezia.apk

06/27/2014 11:56 PM 5,291,956 EpsonPrintServicePlugin_1_0_0.apk

06/27/2014 11:56 PM 5,168,340 CanonPrintService_Kindle.apk

06/27/2014 11:56 PM 5,153,791 com.android.calendar.apk

06/27/2014 11:56 PM 4,587,457 com.android.deskclock.apk

06/27/2014 11:56 PM 4,095,202 com.audible.application.store.apk

06/27/2014 11:56 PM 4,058,844 MapsAPIClientServices-release-signed.apk

06/27/2014 11:56 PM 4,003,205 com.amazon.csapp.apk

06/27/2014 11:56 PM 4,001,940 com.amazon.legalsettings.apk

06/27/2014 11:56 PM 3,689,768 CanonPrintService-KitKat.apk

06/27/2014 11:56 PM 3,221,922 com.amazon.kindle.kso.apk

06/27/2014 11:56 PM 3,030,433 com.goodreads.kindle.apk

06/27/2014 11:56 PM 2,803,672 com.amazon.ags.app.apk

06/27/2014 11:56 PM 2,584,393 com.android.providers.contacts.apk

06/27/2014 11:56 PM 2,529,651 com.android.contacts.apk

06/27/2014 11:56 PM 2,502,292 com.amazon.kindle.otter.settings.apk

06/27/2014 11:56 PM 2,331,108 com.amazon.kindle.otter.oobe.apk

06/27/2014 11:56 PM 2,260,261 com.android.providers.calendar.apk

06/27/2014 11:56 PM 2,202,469 com.amazon.kindle.unifiedSearch.apk

06/27/2014 11:56 PM 2,200,898 com.amazon.videoads.shortvideocontent.apk

06/27/2014 11:56 PM 2,195,108 com.amazon.kindle.rdmdeviceadmin.apk

06/27/2014 11:56 PM 2,107,866 com.amazon.dataplanmanager.apk

06/27/2014 11:56 PM 1,894,085 DeviceSoftwareOTA.apk

06/27/2014 11:56 PM 1,797,728 com.amazon.kor.demo.apk

06/27/2014 11:56 PM 1,758,918 com.amazon.zico.apk

06/27/2014 11:56 PM 1,465,238 WhisperplayCore.apk

06/27/2014 11:56 PM 1,397,061 com.amazon.kindleautomatictimezone.apk

06/27/2014 11:56 PM 1,340,556 RemoteSettingsAndroid.apk

06/27/2014 11:56 PM 1,269,908 com.amazon.webapp.apk

06/27/2014 11:56 PM 1,126,632 DeviceMessagingAndroid.apk

06/27/2014 11:56 PM 998,719 com.amazon.calculator.apk

06/27/2014 11:56 PM 984,997 DeviceBackupAndRestore-release.apk

06/27/2014 11:56 PM 972,096 com.amazon.kindle.devicecontrols.apk

06/27/2014 11:56 PM 967,225 com.amazon.device.sync.sdk.internal.apk

06/27/2014 11:56 PM 894,865 com.amazon.parentalcontrols.apk

06/27/2014 11:56 PM 792,456 com.amazon.weather.service.apk

06/27/2014 11:56 PM 753,066 com.amazon.device.sync.apk

06/27/2014 11:56 PM 713,118 LogManager-1074910.apk

06/27/2014 11:56 PM 698,829 CrashManager-2080810.apk

06/27/2014 11:56 PM 688,758 MobileAuthenticationPlatformAndroid-1001000610.apk

06/27/2014 11:56 PM 678,364 amazon.jackson-19.apk

06/27/2014 11:56 PM 606,792 com.amazon.photos.importer.apk

06/27/2014 11:56 PM 598,486 EpsonPrintPlugin_BuildIn_1_0_5.apk

06/27/2014 11:56 PM 542,892 com.amazon.imp_130139410.apk

06/27/2014 11:56 PM 472,187 PrinterSetup-release.apk

06/27/2014 11:56 PM 463,231 WhisperlinkSdk.apk

06/27/2014 11:56 PM 455,129 com.amazon.dp.contacts.apk

06/27/2014 11:56 PM 358,115 com.amazon.kindle.personal_music.apk

06/27/2014 11:56 PM 348,220 com.amazon.h2settingsfortablet.apk

06/27/2014 11:56 PM 322,939 IvonaTTS.apk

06/27/2014 11:56 PM 304,548 com.amazon.tcomm.apk

06/27/2014 11:56 PM 299,297 HorizonAndroidSourcingService-release.apk

06/27/2014 11:56 PM 286,517 com.amazon.socialplatform.apk

06/27/2014 11:56 PM 278,939 MetricsService-2088210.apk

06/27/2014 11:56 PM 278,840 com.amazon.sharingservice.android.client.proxy.release.apk

06/27/2014 11:56 PM 255,060 com.amazon.alta.h2clientservice.apk

06/27/2014 11:56 PM 191,470 AmazonPlatform-release.apk

06/27/2014 11:56 PM 187,525 com.amazon.h2sharingsettings.apk

06/27/2014 11:56 PM 164,170 FireApplicationCompatibilityEnforcer.apk

06/27/2014 11:56 PM 134,756 com.amazon.communication.apk

06/27/2014 11:56 PM 96,420 com.amazon.dp.fbcontacts.apk

06/27/2014 11:56 PM 78,424 com.amazon.digital.asset.ownership.app.apk

06/27/2014 11:56 PM 75,716 WhisperplayActivityView.apk

06/27/2014 11:56 PM 68,697 DeviceBackupAndRestoreInternalSDK-release.apk

06/27/2014 11:56 PM 58,886 UnifiedShareActivityChooser.apk

06/27/2014 11:56 PM 57,964 com.amazon.cloud9.systembrowserprovider.apk

06/27/2014 11:56 PM 53,576 com.amazon.kindle.identityservice.apk

06/27/2014 11:56 PM 50,949 DeviceClientPlatformContractsFramework.apk

06/27/2014 11:56 PM 49,304 DeviceSoftwareOTAContracts.apk

06/27/2014 11:56 PM 45,715 com.amazon.kindle.personal_video.apk

06/27/2014 11:56 PM 38,032 DeviceMessagingAndroidSDK.apk

06/27/2014 11:56 PM 37,785 DeviceMessagingAndroidInternalSDK.apk

06/27/2014 11:56 PM 34,151 MetricsApi-2053110.apk

06/27/2014 11:56 PM 20,395 RemoteSettingsInternalSDK.apk

06/27/2014 11:56 PM 19,921 com.amazon.unifiedsharefacebook.apk

06/27/2014 11:56 PM 17,466 com.amazon.communication.discovery.apk

06/27/2014 11:56 PM 16,473 com.amazon.unifiedsharesinaweibo.apk

06/27/2014 11:56 PM 14,006 com.amazon.unifiedsharegoodreads.apk

06/27/2014 11:56 PM 13,758 com.amazon.unifiedsharetwitter.apk

06/27/2014 11:56 PM 13,712 MagicWindowAppFireOS_10000010.apk

06/27/2014 11:56 PM 9,576 FireApplicationCompatibilityEnforcerSDK.apk

06/27/2014 11:56 PM 8,849 com.amazon.dp.logger.apk

06/27/2014 11:56 PM 6,603 MapsAPISupportLibrary-release-signed.apk

06/27/2014 11:56 PM 5,999 FireRecessProxy.apk

06/27/2014 11:56 PM 4,962 IdentifiesSDK-release.apk