Do any cells change in size or mass as mammals grow?
I'm wondering if it is safe to assume that the approximate number of cells per unit mass in a mammal will remain fairly constant throughout its lifespan.


Not exactly.  When a tissue is put under stress, it can respond in four main ways:


Hypertrophy - individual cells get larger.  E.g. stressed muscle cells get bigger.
Atrophy - invidivual cells get smaller.  E.g. naturally in the thymus during development.
Hyperplasia - increased cell division to produce more cells.  E.g. mammary gland cells in pregnancy
Metaplasia - one type of specialised cell is replaced with another, usually more durable one, E.g. columnar epithelium in the respiratory tract of a smoker being replaced with squamous epithelium.


The first three responses to stress can change both cell size (auxetic response) and number (multiplicative response), meaning you can't really say that cells per unit mass stays constant throughout life. 

Muscle tissue grows predominantly by hypertrophy, meaning that the muscle gains during puberty are likely to be more as a result of cells getting larger than the development of new ones. 

Both auxetic and multiplicative growth occurs as the body develops.  In addition to this, accretionary growth also occurs - where connective tissues such as bone and cartilage increase in size.