Is it wrong to refer to results as being "highly significant"? Why do statisticians discourage us from referring to results as "highly significant" when the $p$-value is well below the conventional $\alpha$-level of $0.05$?
Is it really wrong to trust a result that has 99.9% chance of not being a Type I error ($p=0.001$) more than a result that only gives you that chance at 99% ($p=0.01$)?
 A: A test is a tool for a black-white decision, i.e. it tries to answer a yes/no question like 'is there a true treatment effect?'. Often, especially if the data set is large, such question is quite a waste of resources. Why asking a binary question if it is possible to get an answer to a quantitative question like 'how large is the true treatment effect?' that implicitly answers also the yes/no question? So instead of answering an uninformative yes/no question with high certainty, we often recommend the use of confidence intervals that contains much more information.
A: This is a common question.
A similar question may be "Why is p<=0.05 considered significant?" (http://www.jerrydallal.com/LHSP/p05.htm)
@Michael-Mayer gave one part of the answer: significance is only one part of the answer.  With enough data, usually some parameters will show up as "significant" (look up Bonferroni correction).  Multiple testing is a specific problem in genetics where large studies looking for significance are common and p-values <10-8 are often required (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2621212/).
Also, one issue with many analyses is that they were opportunistic and not pre-planned (i.e. "If you torture the data enough, nature will always confess." - Ronald Coase).
Generally, if an analysis is pre-planned (with a repeated-analysis correction for statistical power), it can be considered significant.  Often, repeated testing by multiple individuals or groups is the best way to confirm that something works (or not).  And repetition of results is most often the right test for significance.
A: I think there is not much wrong in saying that the results are "highly significant" (even though yes, it is a bit sloppy).
It means that if you had set a much smaller significance level $\alpha$, you would still have judged the results as significant. Or, equivalently, if some of your readers have a much smaller $\alpha$ in mind, then they can still judge your results as significant.
Note that the significance level $\alpha$ is in the eye of the beholder, whereas the $p$-value is (with some caveats) a property of the data.
Observing $p=10^{-10}$ is just not the same as observing $p=0.04$, even though both might be called "significant" by standard conventions of your field ($\alpha=0.05$). Tiny $p$-value means stronger evidence against the null (for those who like Fisher's framework of hypothesis testing); it means that the confidence interval around the effect size will exclude the null value with a larger margin (for those who prefer CIs to $p$-values); it means that the posterior probability of the null will be smaller (for Bayesians with some prior); this is all equivalent and simply means that the findings are more convincing. See Are smaller p-values more convincing? for more discussion.
The term "highly significant" is not precise and does not need to be. It is a subjective expert judgment, similar to observing a surprisingly large effect size and calling it "huge" (or perhaps simply "very large"). There is nothing wrong with using qualitative, subjective descriptions of your data, even in the scientific writing; provided of course, that the objective quantitative analysis is presented as well. 

See also some excellent comments above, +1 to @whuber, @Glen_b, and @COOLSerdash.
