Does increasing the value of k in k-fold coverage requires more memory?
In Next Generation Sequencing (or better called 2nd generation sequencing) you are sequencing many many small genome fragments cut from many DNA molecules. Multiplying molecules, cutting them into fragments and amplifying them is known as library preparation (library of fragments to be read). During that process (of which PCR is important procedure) different errors or biases could be introduced.
What you get from a sequencer is many many short reads in form of FASTQ files that you need to assemble into genome sequence (by de novo assembly or more commonly by alignment to reference genome). It is important to know how sure are we for each nucleotide in assembled genome, which depends on how many times (and with what quality score) sequencer read each nucleotide (see image below).

(source: davidson.edu)
Coverage could mean average coverage across whole genome or coverage of some smaller portion of it (different coverage across genome is usually because of mentioned biases introduced in library prep).
To answer question: In final representation you get one whole genome assembled, if all you need is final representation (so answer 2). If you also store reads, then more coverage means more memory, but final genome is of same length.
NOTE: Coverage is even more ambiguous term that I just described, so you should look at this thread for possible different meanings (for example: which portion of whole genome is covered by sequencing).