Hi kneeddough,
The first place I would start your research is the mattress shopping tutorial here which has all the basic information, steps, and guidelines you will need to make the best possible choice no matter what your budget range or which types of mattresses or materials you tend to prefer.
It would depend to some degree on the specifics of the foam and on the type and degree of compression but in most cases short term compression during shipping or storage (for a few weeks) won’t have any effect on the durability or performance of the foam. There is no such thing as the “best memory foam mattress” … only a memory foam mattress that is “best for you” based on the criteria that are most important to you. A mattress that is the “best match” for someone else may be completely unsuitable for you to sleep on. There is more about the most important parts of the “value” of a mattress purchase in post #13 here.
There is more information in post #9 here about the different ways that one mattress may compare to or “match” another one.
The tutorial post also includes a link to a list of the better online memory foam retailers or manufacturers I’m aware of and many of them sell mattresses that are designed to approximate specific major brand mattresses (usually Tempurpedic because they are the most well known memory foam mattress) and they will say so in their descriptions. In many cases as well … online retailers and manufacturers are very familiar with the Tempurpedic lineup and if you have a more detailed conversation with them on the phone they will be able to help “talk you through” which of their mattresses are closest to the Tempurpedic mattress that your testing indicates is the best match for you in terms of PPP (Posture and alignment, Pressure relief, and Personal preferences). For example Christeli lists their Danieli mattress as being comparable to the Rhapsody Breeze and they list their Briella mattress as being comparable to the Cloud Luxe Breeze.
There may also be some local options that carry mattresses that are similar to the Tempurpedic mattresses but in more reasonable budget ranges as well and if you let me know your city or zip code I’d be happy to let you know about any of the better options or possibilities I’m aware of in your area.
Phoenix