I built a bot from scrap parts from scratch for about £60, but it lost its only fight. Against a DVD player.
It all depends on your technical expertise - if you're good with building, you'll be able to build a very sturdy machine at a fraction of the cost, and if you're not at all skilled like me, you'll probably need to spend a bit more on it to get it about right. The most expensive components usually though are the electronics, but again, it depends what stuff you buy - my hacked 85A ESCs from Botbitz are exceptional, but cost £100, you can buy them for much cheaper elsewhere, but the quality or amp rating might be lower for example. Brushless drive systems are also starting to get a bit of usage from what I've heard - I'm experimenting with a few setups and I'm led to believe there's a few actually in use, but they're not widely used as of yet I don't think.

You mentioned about a rambot, and their strengths are all to do with their toughness and sheer pushing power, so to excel to that, I'd suggest getting 4 drill motors for 4 wheel drive, and the strongest armour you can find, especially for some kind of shovel at the front.
The smaller you can get it, the better too - get as close to the 13.6kg weight limit as you can, but having a huge robot means you'll have to spread the armour more thinly to cover everything. If your robot is half the size, you can have armour twice as thick and that'll help keep the spinners out!

I'd suggest going the cheaper route and using whatever scrap you can find to make the chassis out of etc. just to keep costs down, then you can learn by going to events and fighting, making improvements on top of that and of course have a bit of a cheap thrill! Well, comparatively cheap anyway. Do plan though, planning it out beforehand is a good first step!