Android Applications

IME

The Android system has no built-in support for Japanese input methods. However, a number of apps with this function are available on the Market.

Simeji - Great Japanese IME
HanWriting IME 漢書
MultiLing Keyboard
FlickWnn: to write japanese, but not by drawing
Google日本語入力 (looks like an IME, but I don't know how to use it)
ATOK
Kaede IME Dictionary + Kaede IME Helpfile + Kaede IME User-Dictionary Manager
GO Keyboard + Japanese for GO Keyboard
Swypeis a really neat keyboard replacement that lets you type words by swiping your fingers across letters. Japanese support is downloadable in the settings.
7notes with mazec IME handwriting pad
MyScript Stylus (Beta) IME handwriting pad, Japanese downloadable in the settings
Google Handwriting Input IME handwriting keyboard which supports Japanese

Vertical Text Viewer - Aozora Bunko formatted text reader
Aozora Provider - Makes downloading books from Aozora Bunko easier.
Aoisora
Jade Reader - Open source text file reader with built-in offline Japanese-English dictionary
Dokusho Text file reader with Japanese-English dictionary
OCR Manga Reader - Manga reader with OCR support and support for both EDICT and EPWING dictionaries, made by koohii forum user cb4960. (Related Koohii forum thread)

AnkiDroid - Anki for Android phones - free - Ability to sync with your decks uploaded to AnkiWeb. Includes card editor, review timer. Whiteboard feature to scribble on your screen during reviews. Will support custom fonts from v0.7.
Kanji LS Touch - Good for practicing stroke order.
常用漢字筆順辞典 FREE - Good for practicing stroke order. App all in Japanese.
Kanji Ryokucha - Good for Kanji Koohii SRS reviewing
Browser - to review your Anki decks using the AnkiWeb interface.

TuneIn - Listen to live radio streams from all over the world (pick Japanese stations via "Location" not via "Language" | lots of music streams)
Raziko - Listen to Japanese live streams (there are slightly different stations accessible according to your location setting | mainly spoken words)
RadioJapan - Listen to Japanese live streams)
BeyondPod - Great podcast manager I use to handle my Japanese podcasts
WorkAudioBook - Audiobook player with features for language learners (automatically splits book into phrases, phrase repeat, place saving, subtitle support, dictionary look-up, etc.).

Abduction
sudoku
Bonsai Blast
ANDOROMONA - 2ch viewer
kanji power: An evil game that quizzes the user on kanji that even most Japanese people would be lucky to know.
小学生手書き漢字ドリル1006 - Fun ninja-themed kanji writing game (elementary school kanji version)
中学生漢字(書き取り編) & 中学生漢字(読み方編) - Fun ninja-themed kanji reading/writing games (middle school kanji version)
手書き四字熟語1000 Yojijukugo writing game
Kasui - Tetris-like kanji game
Seppukuman - Hangman using Japanese words

Clipstore - A Clipboard monitoring program that I run in the background. It keeps a list of stuff copied - so when I'm studying something in an Anki deck and I want to mark it for further study later or something, I just copy it and it creates a list for later.
himekuri10: Japanese-style old timey calendar. I mostly use it to get the lunar Calendar date and to amuse myself with who was born on that day.
Feedly - RSS reader which is easy to search and add Japanese RSS feeds from.
Google変換マッシュ (potentially unstable)
Kaki Memo Lite - Good to make a notice by drawing, but no recognition, every character is a picture in the document
OpenVPN Connect -Useful with Japanese VPN (Instructions)
More Locale 2 -To turn the interface language to Japanese (Android 4.2+ Instructions)
Fontomizer SP Add Motoya Japanese fonts to Samsung Galaxy phones