Are there any significant uses of the compound formed by magnesium and anthracene?
It seems like an idea of using magnesium anthracene systems for the $\ce{MgH2}$ production persisted since 1980s [1] till late 2000s, when new more efficient method with better scalability for industrial use was established.

One of the recent reviews in hydrogen-storage applications [2, p. 220] compares the older two-step process of $\ce{MgH2}$ synthesis:


  First, metallic magnesium is dissolved in anthracene to form Mg-anthracene solution. Then, the hydrogenation of the Mg-anthracene solution produces $\ce{MgH2}$ with $\ce{CrCl3}$ or $\ce{TiCl4}$ catalyst. 


with the most recent direct synthesis from Mg powders and tablets [3]. Their experimental trial involved the following:


  Powder and tablet Mg were placed on trays inside a 50 kg hydrogenation furnace. After purging with Ar, the furnace was ﬁlled with 99.9999% hydrogen with a pressure of less than 2 MPa. The temperature of the process was controlled so as to remain near the boundary between the regions $\ce{\{MgH2\}}$ and $\ce{\{Mg + H2\}}$ in the thermal equilibrium diagram, and maintained for 30–45 h. The hydrogen consumption behavior was monitored by a pressure gauge. Afterward, the furnace was gradually cooled down to room temperature. The hydrogenation yield (%) was calculated via the weight difference of Mg samples before/after the process.


$\ce{MgH2}$ obtained with this method showed a hydrogen yield of around 96% with great reproducibility and scalability, and was proved as industry-ready.

But still, magnesium-anthracene · 3THF is used at a lab-scale in fine organic, organometallic, and inorganic syntheses [4, pp. 300-307]:  


as a source of soluble zerovalent magnesium;   
in preparation of finely-divided metal powders and transition metal complexes;   
catalytic synthesis of Grignard compounds.    


Bibliography


Bogdanovic, B. Acc. Chem. Res. 1988, 21 (7), 261–267. DOI 10.1021/ar00151a002.   
Sasaki, K.; Li, H.-W.; Hayashi, A.; Yamabe, J.; Ogura, T.; Lyth, S. M. Hydrogen Energy Engineering: A Japanese Perspective; Springer Japan: Tokyo, 2016. ISBN 978-4-431-56042-5.   
Uesugi, H.; Sugiyama, T.; Nii, H.; Ito, T.; Nakatsugawa, I. Journal of Alloys and Compounds 2011, 509, S650–S653. DOI 10.1016/j.jallcom.2010.11.047.   
Fürstner, A. Active metals: preparation, characterization, applications; VCH: Weinheim; New York, 1996. ISBN 978-3-527-61517-9.