Reference request for a book that covers analytic continuation in great detail starting from basics

The most complete monograph on analytic continuation is perhaps still the more-than-70-years old monograph of Bieberbach [1], which is entirely (and exclusively) devoted to the subject. It deals extensively with various known continuation theorems, which give (usually sufficient) conditions for a power series to be continuable across the boundary of its convergence disk (including a necessary and sufficient condition for a point to be non-regular, i.e. to be a point across which it cannot be analytically continued), and with almost all the known methods for analytically continuing power series (including but not limiting to Borel-Laplace and Laplace transforms). Also it gives very detailed historical context for the development of the subject. However, it has a few evident defects namely

1. It is written in German (and this perhaps is not really a serious flaw) and
2. It is outdated in some of his sections (time has passed even for this great classic), and finally (perhaps the worst one)
3. It does assume the basic concepts are known, thus it does not develope the topic from scratch.

Said that, a treatment that is not as comprehensive but builds up from the basics up to intermediate levels of difficulty and offers also several exaples and exercises, can be found in the books of Markushevich, particularly [3], chapter 9 ("Analytic continuation"), but also [2], chapters 16 and 17 ("Power series: rudiments" and "Power series: ramifications") of volume I and chapter 8 ("Analytic continuation") of volume III. The exposition starts from basic conceps i.e. singularities of analytic functions, the convergence disk of a power series (including the same necessary and sufficient condition for a point to be non-regular given by in [1]), the meaning of an analytic element, etc. and goes up to giving an elementary introduction to the Borel-Laplace transform.

My advice

My advice is to use the textbooks of Markushevich to get aquainted to the topic at a deeper level than other textbooks allow, and use the book of Bieberbach as a sourcebook when you need to go to further deeper. Indeed, even if you don't understand German very well, you can use [1] as a guide to the many English or French language original papers: I follow this route almost always.

References

[1] Ludwig Bieberbach, Analytische Fortsetzung (German) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 3, Berlin-Göttingen-Heidelberg: Springer-Verlag, pp. IV+168 (1955), MR0068621, Zbl 0064.06902.

[2] Alekseĭ Ivanovich Markushevich, Theory of functions of a complex variable. Vol. I. (English), Selected Russian Publications in the Mathematical Sciences, Englewood Cliffs, N.J.: Prentice-Hall, Inc., pp. XIV+459 (1965), MR0171899, Zbl 0135.12002, and
Theory of functions of a complex variable, Vol. III. (English), Selected Russian Publications in the Mathematical Sciences, Englewood Cliffs, N.J.: Prentice-Hall, Inc., pp. xi+360 (1967), MR0215964, Zbl 0148.05201.

[3] Alekseĭ Ivanovich Markushevich, The theory of analytic functions: a brief course, translated from the Russian by Eugene Yankovsky (revised from the 1978 Russian edition) (English), Moscow: Mir Publishers. pp. 423 (1983), MR0708893, Zbl 0499.30002.