have read in a few places such as in the book Shaarei Kedusha that God is "infinite" (Ein Sof) (without end). What does this mean? Typically we think of something infinite as being immeasurable or mathematically as the limit of a function as some variable increases indefinitely. The latter does not apply to God since He does not change (Rambam Yesodei Torah 1). What then does infinite mean in this context? (note that I am not asking about God's essence as that is forbidden to inquire about. Just asking what we can know (if anything) of what infinite means with regard to Him)
Since we are finite beings, then in this sense infinite means immeasurable or not understandable. Any attempt at description is a limit to something that has no limit. Your statement
Typically we think of something infinite as being immeasurable or mathematically as the limit of a function as some variable increases indefinitely.
is itself an attempt to describe the indescribeable. Thus we say that looking at a limited subset of characteristics that we ascribe to Hashem causes us to be at a point and move along.
Ein Sof, while it translates as infinite has more the connotation of unable to be expressed, as any expression implies some sort of limit or boundary.
For example, an exponential function in mathematics may be infinite, but since it approaches a limit without bound, it is not Ein Sof.
Another example is Aleph number which are infinite values but are also not Ein Sof as the bounds can be explicitly expressed.
I see two questions here: how is Hashem infinite and what does Ein Sof mean
Infinite according to Google means:
limitless or endless in space, extent, or size; impossible to measure or calculate. "the infinite mercy of God" synonyms: boundless, unbounded, unlimited, limitless, never-ending, interminable;
Obviously the terms referring to physicality don't apply, but the other ones are appropriate.
Lehavdil, Ein Sof as explained by Nefesh HaChaim Shaar Beis Chapter 2
מה שבז''הק מכנהו ית' בשם אין סוף איננו כנוי עליו ית''ש אלא הכוונה על השגתנו אותו מצד כחות הנשפעים מאתו בהתחברותו ברצונו להעולמות. ולזאת כנוהו א''ס ולא אין ראשית. כי באמת מצד עצמותו ית''ש אין לו לא סוף ולא ראשית. רק מצד השגתינו כחותיו ית'. הלא כל השגתינו הוא רק ראשית. אבל אין סוף להגיע בהשגה להשיג את כחותיו ית' הנשפעים
That which the Zohar calls Him Ein Sof (lit: without end), it's not that we're actually using it to give a description of Him. Rather the intention is based on our understanding of Him, based on the powers that have an affect from Him due to His desired connection to the universes. This why we refer to him as Ein Sof and not Ein Reishis (without beginning), because in truth, regarding His Essence, there's no end and no beginning. Only with regards to our comprehension of His powers, they have a beginning, yet they don't have an end. We'll never fully understand His influential powers.
To us, Infinity means forever expanding. But that is because nothing that we relate to is actually infinite. Therefore when we use the term we are referring to the theoretical idea that we can keep on going. For example, when we say that numbers are infinite that doesn't mean that there is a
forever number. It means that there is no foreseeable limit hindering us from continuing to count. Hence, infinity is more a function than a number.
When we say that God is infinite that means that he is indeed forever great. If we were to try to measure Him we would keep on going.
This is comparable to the vastness and endlessness of the athiest`s Original Nothing.