1. A bar fridge to celebrate good fortune
2. A couch to lay down on when you've had a shocker.
EDITED (darkydog beat me to it - ugh)
Seriously, all the best Merriguy.
You say that space is small and budget is not massive, so I'd keep it simple.
As suggested, I would get a dual monitor setup.
One for odds watching and one for the bet placement page.
I wouldn't get a tv card and run that simultaneously with your other connections, you'll eat into your pc resources too much, I'd get foxtel on a flat panel tv and wall mount it above the twin monitors.
Get yourself a good expensive comfortable chair, it makes all the difference.
Have the top of the monitors at eye level
Have two drives in your pc, and use a cloning program to do a daily clone of your existing hard drive, that way if a failure occurs you can just switch drives.
You don't need 17 monitors or anything silly like that, although it looks impressive, it is a waste of resources and expensive.
There are plenty of racing sites that have programs that will display all the odds everywhere for you to choose from or see fluctuations in one browser. (I use dynamicodds).
Enjoy your retirement.
To answer your question directly I can't do without the internet, Excel, and a Captain Kirk leather chair, the rest is just frills.