The experiments are both valuable and tell you different things. The first tells you about the error of your measurement method, including the way the sample is prepared for measurement. Ideally if you measure the same sample three times, you will get the same value, but it is rarely the case. There's always some variation. The second would tell you more about the systematic variation around what you're actually studying. Typically the second experiment is more scientifically interesting and is what you would present at a meeting. The first experiment is part of your method development and would only really be scientifically interesting if you were presenting a new analytical technique. For analysis many instruments implicitly build multiple replicates into the analysis routine and present an average value anyway. If not, measuring the sample multiple times isn't a bad thing to do, but the error associated with multiple measurements of the same sample is almost always much smaller than the variation around multiple independent samples. So I would generally say that once you have developed your method and satisfactorily demonstrated its precision within certain tolerances, measuring multiple completely independent samples is the approach you would want to take.