The following is an extract from a question on Dogster where the qustion raised was "Can eating carrots improve your dogs eyesight?"

Quote :

There is, however, such a thing as having too much vitamin A in the diet. Dogs that have too much in their diet (hypervitaminosis) can develop bone problems and muscle weakness. Thankfully, reaching a toxic level of vitamin A would require a very high dose over a long period of time, and giving your dog a few carrots now and again isn’t going to come close to providing an overdose. If you do choose to give your dog carrots as an occasional treat, it's best to cut them up into small enough pieces to reduce the risk of choking or gastrointestinal discomfort.

... and was the only thing I saw that indicated any negative to feeding your dog carrots.  Of course, eating too much anything may cause problems:

1. Too many carrots may cause a change in your dogs color ( for real! )
   
2. The amount of dietary fiber may produce other results that you might find unnaceptable
   

However, I think it safe to say that the number of carrots to produce a change in color is ridiculously high and you'll just have to experiment to see how your dog handles the fiber content.
Remember that while dogs, unlike cats, can digest some vegetables that they'll gain little nutritive value from the vegetables. Vegans who try to change their dogs diet to a vegetarian one are doing their pets a major disservice - imho!
So, to answer your specific question, a handful of "baby carrots" or a handful of cut up carrot(s) shouldn't be a problem, even every day - now a few bags of carrots a day for years just might be.

Leo loves apples, i usually share some with him from time to time, just cant resist that face. i did use frozen carrots when he was teething.

Awesome, thanks for the replies all - very helpful. I will vary as to not exaggerate with just one type of treat - which is why I tried carrot in the first place actually. Will try apples and watermelon in small pieces next.

just make sure that there are no seeds in the watermelon. Leo and Joey love watermelon too.