The model needs at least one participant in each group and more participant than there are coefficients to be estimated. This means for instance for a one-way ANOVA on k groups you will need at least k+1 participants.

This is the technical minimum requirement. How many participants you will need to get results that are interpretable for you with sufficient confidence depends on the context and on your needs. If you have no idea, then check what is feasable and use as many participants as you can get/handle.

Two. With two samples you can calculate a mean and standard deviation. With one sample, the (n-1) denominator results in division by zero. Not good. At n=2, power will be low (probability of finding a significant difference when it exists). Precision will be low (wide confidence intervals). Repeatability will be problematic (another researcher that does your experiment gets contradictory results). If statistical considerations are the sole factor in determining replication then use as many replicates as you can. Other factors might be cost, time, ethics, risk, and rarity.

A realistic minimum sample size is more difficult. Look at the literature. What do others use? Alternatively, can you simulate your experimental design? The simulation may not be accurate or publishable, but it will help you understand how your choice in sample size will affect your outcome. For simple experiments G*Power might be a simpler way to the same end (http://gpower.hhu.de/). Think of the sample size calculator as an independent study tool to help understand issues with sample size.

Hello Janaa, I believe you need to have response from at least three replicates in each group to be comfortable and confident on your conclusion. Three replicates in each group help you to better understand the COV in your sample. Moreover, you can easily see if there is any cook's point (if any data point is too off from other two) in your responses.   

Three replicates is not a real answer unless your expected outcome is so large that statistics is mostly irrelevant. If the expected variability is one quarter of the mean and you are expecting an order of magnitude difference in means then three replicates might work well. The variability is small relative to the expected effect size.

In more typical cases three replicates gives you a few random numbers and you get another artifact to add to the scientific literature. A part of the answer also depends on the type of experiment. Three replicates at each of 7 levels in a linear regression problem might be very good (especially if the relationship is linear or curvilinear). Three replicates at each of 7 treatments that will then be analyzed using Tukey's HSD test will give you random junk.

The answer should also consider the risk you accept for making a mistake. Is there a chance of litigation if you say one thing and people are harmed because you got it wrong? Also consider what you would want. If this is a clinical trial for a new drug, would you be happy taking the new drug knowing that the experiment used to demonstrate efficacy used 3 replicates. There were no observed side effects in those three replicates, therefore it is safe for you to use?

Even if the outcome is not so critical, as in field plot trials, you should really ask: if it was only worth three replicates was it worth doing in the first place? 

Jochen answered the question as written, but if you are interested in the power (rather than just the minimum for the procedure) and have an ANOVA and want to compare multiple groups in several ways you will need to estimate power for multiple hypotheses. The attached may be for interest for this.

Technical Report Power Analysis for Studies with Multiple Hypotheses

Thank you so much for the inputs! Im actually planning on doing a purposive sampling that's why I asked the minimum for a group

Purposive versus random sampling will not influence sample size. Purposive sampling is great when used to insure that you include specific groups that might be missed by random sampling. Random sampling is great to minimize the possibility that your results are biased by some known or unknown factor. In an ideal world you use a blend of these strategies.

Purposive sampling could influence sample size by increasing the number of "treatments". You might want a sample of 50 surveys of people that enter the local grocery store under a random sampling approach. With purposive sampling you might decide that you want to make sure that you sample from the three dominant cultures/races in your area. This is great because you gain a more balanced insight into the people at the grocery store, but you also need to collect more samples. Given the variability in each sub-population, and the required accuracy of your estimates, you can work out the required sample size. Note that your population size also changes. As you fragment the population more you will increase the total number of surveys that you will need. However, if 50 was the right number for a random sample then I would expect that each subpopulation will not need quite that many. So for three subpopulations you will need something less than 150 surveys. The exact number depends on how much of the initial variability is accounted for by identifying the subpopulations.

I have read of this equation somewhere wherein if you have an unknown population, your smaple size could be solved using 50 + 8 x n where n is the number of dimensions in your scale.