How many unit squares can you pack into a rectangle with nearly integer side lengths?

There is a surprising asymptotic result in 2 dimensions, closely related to your question. See [MR0370368] Erdős, P.; Graham, R. L. On packing squares with equal squares. J. Combinatorial Theory Ser. A 19 (1975), 119–123.

There's a fair bit of literature about efficient packing of squares in squares. See, e.g., Kearney and Shiu, Efficient packing of unit squares in a square, available at http://www.emis.de/journals/EJC/Volume_9/PDF/v9i1r14.pdf, also Walter Stromquist, Packing 10 or 11 unit squares in a square, available at http://www.combinatorics.org/Volume_10/PDF/v10i1r8.pdf

https://erich-friedman.github.io/mathmagic/0610.html has some answers for small numbers of squares.