There are only two limitations on the number of life insurance policies you can have: What you are able to afford, and what the insurance companies deem to be an insurable value. You probably won't need a lot of whole life insurance policies, but specific term life policies can be used to cover expenses related to specific financial obligations. Insurance companies won't underwrite policies that are obviously intended to defraud the insurance system, but everyone is encouraged to insure their full personal value.
For common life insurance needs, such as the home mortgage or making sure that the money will be there for your children's education, term life insurance policies are a better idea than whole life. Term life policies are written for specific cash values, and have a set expiration, such as the final date of your mortgage payments matching the expiration of a term life policy for the value of the home mortgage. This type of policy, because it is not a guaranteed payout, is more affordable than whole, or permanent, life insurance. You can have as many term life policies as you can afford, but keep in mind that you will lose the money invested in term life policies if you outlive the term of the policy.
Whole life insurance does not expire as long as the policy premiums are up to date. It has a guaranteed cash payout amount, and is most often used to leave an inheritance, pay your outstanding debts, and cover funeral and other final expenses. Whole life insurance costs more than term life, but the face value of the policy is a guaranteed payout, regardless of how long you live. There is no set limit on the number of whole life insurance policies you may have, but insurance companies will be looking at your net worth if you apply for multiple policies or inordinate sums of coverage.
When you apply for an insurance company, you are required to list any other life insurance policies you have as well. Insurance companies then compare your total assets with the insurable amount requested to make their decision. You may find it difficult to get insured for much more than your total net worth, and insurance companies that suspect you are attempting to accumulate policies to generate artificial wealth may refuse to offer you any coverage at all. If you can afford it, take out a policy on everything that needs financial protection, but do not over-extend yourself or attempt to claim more insurable value than you can demonstrate.