Below are a list of programs that can all be used over an SSH connection with no need for an X connection.

• Email
• RSS
• Web
• Text Editing
• Spreadsheet/math/calculator
• Calendar
• To-do management
• Music/audio
• Chat
• Database
• Filesystem
• Version control
• Addressbook
• Games
• Torrents
• Admin
• Miscellaneous

Email:

• mail (command-line; uses local mailboxes)
• mutt (full-screen; uses local mailboxes, POP3, or IMAP)
• elm (full-screen; uses local mailboxes, may support POP3/IMAP)
• mh/nmh (mail-handler/new-mail-handler; command-line; uses "MH" format, supports POP3 via fetchmail)
• pine/alpine (full-screen; uses local mailboxes, POP3, or IMAP)
• offlineimap
• and many others

RSS:

• Raggle
• newsbeuter
• snownews
• NRSS/Canto
• rss2email/rss2imap
• Google Reader supports an on-line reader (the Mobile version should be usable in any console browser)

Web:

• lynx (the classic full-screen text-browser)
• links/links2 (a full-screen text-browser with more visual layout engine)
• elinks (a full-screen text-browser with more visual layout engine)
• w3m
• edbrowse (it's an editor, it's a browser, it's command-line)

Text Editing:

• vi/vim (the classic full-screen text-editor)
• emacs (another popular and extensible choice in full-screen text-editors)
• ed (the classic command-line text editor)
• nano (a simple full-screen editor)
• pico (a simple full-screen editor)
• edbrowse (it's like ed on steroids)
• and countless others

these can be used in concert with various markup syntax such as Markdown, HTML, DocBook, LaTeX, etc to produce publishable documents; you can use packages like antiword or wordview ("wv") to convert .DOC files to a usable format.

Spreadsheet/math/calculator:

• Spreadsheet:
  • sc (a simple full-screen spreadsheet with vi-like keybindings)
  • sc-im (like sc above but with the ability to import/export xls/xlsx files)
  • slsc (based on sc)
  • oleo (a simple full-screen spreadsheet with a more Emacs-like feel)
• Math:
  • Octave
  • R
  • scilab
• Graphing:
  • gnuplot
  • graphviz
• Calculator
  • bc (a simple command-line calculator)
  • python (the full power of Python, at a command-line)

Calendar:

• calendar (show events on given days)
• remind (like the previous calendar program on steroids)
• cal (display a calendar)
• pcal (good for printing)
• cron (for scheduling repeated tasks)
• at (for scheduling a single job sometime in the future)
• gcalcli (interact with Google Calendar from the command-line)
• khal
• mencal and mencal2 (menstruation calendars)

To-do/time management:

devtodo

TaskWarrior

TimeTracker (a simple command-line time-tracker in the spirit of many VCS tools, written by yours-truely in response to this post, and somewhat documented here)

Music/audio:

• Playback
  • cmus
  • mpg321
  • mpd/mpc
  • mpg123
  • ogg123
  • pianobar (command-line Pandora client)
  • nama (music recording and editing)
  • and many others
• Editing/recording
  • sox
  • ecasound
• Volume control
  • aumix
• Podcatchers
  • hpodder

Chat:

• finch (a console version of Gaim/Pidgin)
• centericq (support for ICQ, Yahoo!, AIM, IRC, MSN, Gadu-Gadu and Jabber protocols)
• naim (support for AIM, ICQ, IRC, and the lily CMC protocols)
• irssi (popular IRC client)
• weechat
• gtmess (MSN client)

Database:

• psql interface to PostgreSQL
• mysql interface to MySQL
• sqlite interface to the sqlite database
• all the major databases have command-line clients

Filesystem

• Midnight Commander (mc)
• ranger

Version control:

I currently use git, Mercurial, Bazaar, Subversion and RCS depending on the project context

• Distributed VCS
  • git (a fast, powerful patchwork of scripts and commands)
  • mercurial (hg) (fast, powerful, predictable, and mostly written in Python)
  • bazaar (fast, powerful, predictable, and purely written in Python)
  • darcs (slower, and less popular)
• Centralized VCS
  • subversion (svn) (removes some of the annoyances of CVS and more extensible)
  • cvs/OpenCVS (the classic, rapidly being replaced by Subversion)
  • rcs (not bad for one developer and one text file, but doesn't scale nicely)

Addressbook:

• abook
• rolo
• khard

Games:

• the bsdgames package in Debian provides several (I'm a sucker for cribbage)
• the frotz package gives access to most text-adventures
• nethack
• and oodles of other games

Torrents:

rtorrent

aria2c

I also find that using "tmux" or GNU "screen" vital to being productive, as I can do many, many things all at the same time, each in their own window. It also allows me to disconnect and then reconnect from another machine later, resuming where I left off.