What's interesting or not is something that should be decided based on what you know about the topic. It's perfectly fine to run a 2x2 experiment, use only a single manipulation or even study the distribution of a single variable (with no experimental manipulation or bivariate statistics at all) if that variable present some theoretical or practical interest.
One issue here is that you seem to be overly concerned about significance testing. In fact, it's often not reasonable to expect any effect to be exactly zero (this is obviously true and has been stressed many times in relation to observational research – modeling in economics, sociology, political science, etc. – but it is also true for psychology experiments although perhaps not always in chemistry of physics).
It's also incorrect to consider that failing to reject the null hypothesis is strong evidence for the absence of any effect as the power to detect a given effect depends on the sample size and the actual magnitude of this effect. So if you actually expect some variable not to have an effect and you are specifically interested in that, just running an ANOVA in the hope that the p-values are above the usual threshold is not the right strategy, no matter how many factors you have in your design.
For all these reasons, your adviser has a point in the sense that blindly running a 2x2 experiment and finding that no effect is significant might very well leave you with uninterpretable and unpublishable data. Adding some manipulation that appear to “cancel” an effect (as shown by a simple effect and a significant interaction) could be a way to “sneak in” the story about the absence of the effect under some condition but the fact remains that a non-significant effect alone is difficult to interpret and next to impossible to publish in some journals/disciplines.
However, I think the solution here is not necessarily to add a few other factors but to think hard about effect size and formulate more sophisticated hypothesis than “something or other has an effect”. You can then use power analysis, equivalence testing, or precision in parameter estimates to ensure that your experiment provides valuable information on this hypothesis in any case.
Also, in some fields like psychology, researchers work very hard to bring everything to a 2x2x… ANOVA design even when some variables naturally have a quantitative or continuous interpretation. Another way to make an experiment more informative is to use several levels for such independent variables or try to directly model the relationship between these quantitative variables and your response.
Finally, one consideration would be the cost of additional conditions. In survey research, common wisdom is that people don't mind long questionnaires, the difficulty is getting them to participate at all. In that case, researchers often add variables/questions “just in case”, to address ancillary hypotheses or get more out of the effort. At the other extreme, if the study is necessarily between-subject and adding participants is very costly (say brain imaging with hard-to-recruit patients), you would need to think very carefully before adding any factor or condition.